MATH 447: Real Variables — Lecture Notes

Compiled by Jerich Lee

December 08, 2024

Peano Axioms

[Peano Axioms] The natural numbers $ \mathbb{N} $ are defined by the following postulates:

1. $ \mathbb{N} $ contains a distinguished element $ 1 $.
2. Every $ n \in \mathbb{N} $ has its successor in $ \mathbb{N} $, denoted $ S(n) $.
3. $ 1 $ is not the successor of any element in $ \mathbb{N} $.
4. If $ m $ and $ n $ have the same successor, then $ m = n $.
5. If $ A \subseteq \mathbb{N} $ such that $ 1 \in A $ and $ S(n) \in A $ whenever $ n \in A $, then $ A = \mathbb{N} $.

[Uniqueness of $ \mathbb{N} $] Suppose $ X $ is a set with a distinguished element $ 1' $ and a successor map $ S' $, satisfying the Peano Axioms (N1-N5). Then there exists a bijection $ \Phi{} : \mathbb{N} \to{} X $ such that: $$\begin{align*} \Phi(1) = 1', \quad \Phi(S(n)) = S'(\Phi(n)) \, \forall n \in \mathbb{N}. \end{align*}$$

Mathematical Induction

[Principle of Mathematical Induction] Suppose $ (P_n)_{n \in{} \mathbb{N}} $ is a sequence of statements such that:

1. $ P_1 $ is true.
2. For any $ n \in \mathbb{N} $, $ P_n $ implies $ P_{n+1} $.

Then $ P_n $ is true for all $ n \in{} \mathbb{N} $. [Induction Proof] Prove $ \sum_{k=1}^n k = \frac{n(n+1)}{2} $ for $ n \in{} \mathbb{N} $. {Proof.} Base case ($ n=1 $): $$\begin{align} \sum_{k=1}^1 k = 1 = \frac{1(1+1)}{2}. \end{align}$$ Inductive step: Assume $ \sum_{k=1}^n k = \frac{n(n+1)}{2} $. Then: $$\begin{align} \sum_{k=1}^{n+1} k = \sum_{k=1}^n k + (n+1) = \frac{n(n+1)}{2} + (n+1). \end{align}$$ Simplify: $$\begin{align} \sum_{k=1}^{n+1} k = \frac{n(n+1) + 2(n+1)}{2} = \frac{(n+1)(n+2)}{2}. \end{align}$$ Thus, $ \sum_{k=1}^n k = \frac{n(n+1)}{2} $ holds for all $ n $.

Properties of Integers

[Addition Properties of $ \mathbb{Z} $] The integers $ \mathbb{Z} $ satisfy:

1. {Associativity:} $ a + (b + c) = (a + b) + c $ for all $ a, b, c \in \mathbb{Z} $.
2. {Commutativity:} $ a + b = b + a $ for all $ a, b \in \mathbb{Z} $.
3. {Neutral Element:} $ \exists 0 \in \mathbb{Z} $ such that $ a + 0 = a $.
4. {Existence of Opposites:} For every $ a \in \mathbb{Z} $, $ \exists -a \in \mathbb{Z} $ such that $ a + (-a) = 0 $.

[Uniqueness of Additive Elements]

1. The neutral element $ 0 $ is unique.
2. For any $ a \in \mathbb{Z} $, the opposite $ -a $ is unique.

[Addition Properties of $ \mathbb{Z} $]

1. {Associativity:} $ a + (b + c) = (a + b) + c $ for all $ a, b, c \in \mathbb{Z} $.
2. {Commutativity:} $ a + b = b + a $ for all $ a, b \in \mathbb{Z} $.
3. {Neutral Element:} $ \exists 0 \in \mathbb{Z} $ such that $ a + 0 = a $ for all $ a \in \mathbb{Z} $.
4. {Existence of Opposites:} $ \forall a \in \mathbb{Z}, \exists -a \in \mathbb{Z} $ such that $ a + (-a) = 0 $.

[Uniqueness of $ 0 $ and $ -a $]

1. The neutral element $ 0 $ is unique.
2. For each $ a \in \mathbb{Z} $, the opposite $ -a $ is unique.

[Multiplication Properties of $ \mathbb{Z} $]

1. {Associativity:} $ a \cdot (b \cdot c) = (a \cdot b) \cdot c $ for all $ a, b, c \in \mathbb{Z} $.
2. {Commutativity:} $ a \cdot b = b \cdot a $ for all $ a, b \in \mathbb{Z} $.
3. {Neutral Element:} $ \exists 1 \in \mathbb{Z} $ such that $ 1 \cdot a = a $ for all $ a \in \mathbb{Z} $.
4. {Distributive Law:} $ (a + b) \cdot c = a \cdot c + b \cdot c $ for all $ a, b, c \in \mathbb{Z} $.

[Multiplication by Zero] For any $ a \in{} \mathbb{Z} $, $ 0 \cdot{} a = 0 $. [Field Properties of $ \mathbb{Q} $] The rational numbers $ \mathbb{Q} $ satisfy:

1. {Inverse:} $ \forall a \in \mathbb{Q} \setminus \{0\}, \exists a^{-1} \in \mathbb{Q} $ such that $ a \cdot a^{-1} = 1 $.

If $ (X, +, 0, \cdot{}, 1) $ satisfies (A1–A4), (M1–M4), and the distributive law, $ X $ is called a field.

Ordered Fields

[Ordered Fields] A field $ F $ is ordered if equipped with a linear order $ \leq{} $ such that:

1. If $ a \leq b $, then $ a + c \leq b + c $ for all $ a, b, c \in F $.
2. If $ a \leq b $ and $ c \geq 0 $, then $ ac \leq bc $.

[Properties of Ordered Fields] Let $ F $ be an ordered field. Then for all $ a, b, c \in{} F $:

1. If $ a \leq b $, then $ -b \leq -a $.
2. If $ a \leq b $ and $ c \leq 0 $, then $ bc \leq ac $.
3. If $ 0 \leq a $ and $ 0 \leq b $, then $ 0 \leq ab $.
4. $ 0 \leq a^2 $ for all $ a \in F $.

Rational Zeros Theorem

[Rational Zeros Theorem] Suppose $ p(x) = c_nx^n + \ldots{} + c_1x + c_0 $, with $ c_0, \ldots{}, c_n \in{} \mathbb{Z} $, $ c_0 \neq{} 0 $, $ c_n \neq{} 0 $. If $ p(r) = 0 $ for $ r = \frac{c}{d} $ (where $ c, d \in{} \mathbb{Z} $, $ d \neq{} 0 $, $ \gcd{}(c, d) = 1 $), then $ c \mid{} c_0 $ and $ d \mid{} c_n $. [Irrationality of $ \sqrt{2} $] No rational number $ r $ satisfies $ r^2 = 2 $.

Ordered Fields and Completeness

Fields and Order

If $ F $ is a field with more than one element, then $ 0 \neq{} 1 $.
[Ordered Fields] A field $ F $ is called ordered if it is equipped with a linear order $ \leq{} $ satisfying:

1. If $ a \leq b $, then $ a + c \leq b + c $ for all $ a, b, c \in F $.
2. If $ a \leq b $ and $ c \geq 0 $, then $ ac \leq bc $.

Properties of Ordered Fields

Let $ F $ be an ordered field. Then for all $ a, b, c \in{} F $:

1. If $ a \leq b $, then $ -b \leq -a $.
2. If $ a \leq b $ and $ c \leq 0 $, then $ bc \leq ac $.
3. If $ 0 \leq a $ and $ 0 \leq b $, then $ 0 \leq ab $.
4. $ 0 \leq a^2 $ for all $ a \in F $.

Absolute Value and Distance

[Absolute Value] For $ a \in{} F $, the absolute value $ |a| $ is defined as: $$\begin{align} |a| = \begin{cases} a & \text{if } a \geq 0, \\ -a & \text{if } a < 0. \end{cases} \end{align}$$ [Distance] The distance between $ a, b \in{} F $ is defined as: $$\begin{align} \text{dist}(a, b) = |a - b|. \end{align}$$

Completeness Axiom

[Completeness Axiom] If $ S \subset{} \mathbb{R} $ is non-empty and bounded above, then it has a unique least upper bound (supremum), denoted $ \sup{} S $. [Archimedean Property] If $ a, b > 0 $ in $ \mathbb{R} $, then there exists $ n \in{} \mathbb{N} $ such that $ n \cdot{} a > b $. [Denseness of $ \mathbb{Q} $] The rational numbers $ \mathbb{Q} $ are dense in $ \mathbb{R} $, meaning that for any $ a, b \in{} \mathbb{R} $ with $ a < b $, there exists $ r \in{} \mathbb{Q} $ such that $ a < r < b $.

Sequences and Limits (Sections 7-9)

Definitions and Examples

[Sequence] A sequence is a function $ s : \{m, m + 1, \dots{} \} \to{} \mathbb{R} $ (for $ m \in{} \mathbb{Z} $). We denote the sequence as $ (s_n)_{n \geq{} m} $, where $ s_n = s(n) $. [Convergence and Limit] A sequence $ (s_n) $ converges to $ L \in{} \mathbb{R} $ if: $$\begin{align} \forall \varepsilon > 0, \exists N \in \mathbb{R} \text{ such that } |s_n - L| < \varepsilon \text{ for } n > N. \end{align}$$ We write $ \lim_{n \to \infty} s_n = L $ or $ s_n \to{} L $. [Uniqueness of Limits] A sequence cannot have more than one limit.

Examples of Limits

1. $ \lim_{n \to \infty} \frac{1}{n} = 0 $.
2. The sequence $ ((-1)^n)_{n \in \mathbb{N}} $ does not converge.
3. $ \lim_{n \to \infty} \frac{3n + 1}{2n + 1} = \frac{3}{2} $.

Facts about Limits

If $ (s_n) $ converges, and $ s_n \geq{} a $ for all but finitely many $ n $, then $ \lim{} s_n \geq{} a $. If $ s_n \geq{} 0 $ for all $ n $ and $ \lim{} s_n = s $, then $ \lim{} \sqrt{s_n} = \sqrt{s} $.

Convergent Sequences are Bounded

[Bounded Sequence] A sequence $ (s_n) $ is called bounded if $ \exists{} A \in{} \mathbb{R} $ such that $ |s_n| \leq{} A $ for all $ n $. Convergent sequences are bounded.

Arithmetic of Limits

[Limits of Sums, Products, and Ratios] Suppose $ \lim{} s_n = s $ and $ \lim{} t_n = t $. Then:

1. $ \lim (s_n + t_n) = s + t $,
2. $ \lim (a \cdot s_n) = a \cdot s $ for any $ a \in \mathbb{R} $,
3. $ \lim (s_n \cdot t_n) = s \cdot t $,
4. If $ t \neq 0 $, then $ \lim \frac{s_n}{t_n} = \frac{s}{t} $.

Sums, Products, Ratios of Limits (Section 9)

Arithmetic of Limits

[Arithmetic of Limits] Suppose $ \lim{} s_n = s $ and $ \lim{} t_n = t $. Then:

1. $ \lim (s_n + t_n) = s + t $,
2. $ \lim (a \cdot s_n) = a \cdot s $ for any $ a \in \mathbb{R} $,
3. $ \lim (s_n \cdot t_n) = s \cdot t $,
4. If $ t \neq 0 $, then $ \lim \frac{s_n}{t_n} = \frac{s}{t} $.

[Squeeze Theorem] If $ a_n \leq{} s_n \leq{} b_n $ for all $ n $, and $ \lim{} a_n = \lim{} b_n = s $, then $ \lim{} s_n = s $.

Basic Examples of Limits

[Basic Examples]

1. $ \lim \frac{1}{n^p} = 0 $ for $ p > 0 $,
2. $ \lim a^n = 0 $ if $ |a| < 1 $,
3. $ \lim n^{1/n} = 1 $,
4. $ \lim a^{1/n} = 1 $ for $ a > 0 $.

Diverging Sequences

[Divergence to Infinity] We say $ \lim{} s_n = +\infty{} $ if for all $ A > 0 $, there exists $ N \in{} \mathbb{R} $ such that $ s_n > A $ for $ n > N $. Similarly, $ \lim{} s_n = -\infty{} $ is defined. [Product Rule for Divergence] If $ \lim{} s_n = +\infty{} $ and $ \lim{} t_n > 0 $, then $ \lim{} (s_n \cdot{} t_n) = +\infty{} $. If $ s_n > 0 $ for all $ n $, then $ \lim{} s_n = +\infty{} $ if and only if $ \lim{} \frac{1}{s_n} = 0 $.

Monotone Sequences (Section 10)

[Monotone Sequences] A sequence $ (s_n) $ is:

1. \emph{Increasing} if $ s_n \leq s_{n+1} $ for all $ n $,
2. \emph{Decreasing} if $ s_n \geq s_{n+1} $ for all $ n $,
3. \emph{Monotone} if it is either increasing or decreasing.

[Examples of Monotone Sequences]

1. $ x_n = \sum_{k=1}^n \frac{1}{k^2} $ is increasing because $ x_{n+1} = x_n + \frac{1}{(n+1)^2} > x_n $.
2. $ y_n = \frac{(-1)^n}{n^2} $ is not monotone because $ y_{n+1} > y_n $ if $ n $ is odd, and $ y_{n+1} < y_n $ if $ n $ is even.

Monotone Sequences and Convergence (Section 10)

Monotone Sequences

[Monotone Sequences] A sequence $ (s_n) $ is:

1. \emph{Increasing} if $ s_n \leq s_{n+1} $ for all $ n $,
2. \emph{Decreasing} if $ s_n \geq s_{n+1} $ for all $ n $,
3. \emph{Monotone} if it is either increasing or decreasing.

[Theorem 10.2] Any monotone bounded sequence converges. [Bounded Monotone Sequence] Consider $ s_n = \sum_{k=0}^{n-1} \frac{1}{k!} $. This sequence is:

1. Increasing, since $ s_{n+1} = s_n + \frac{1}{n!} > s_n $.
2. Bounded, since $ s_n \leq 3 $ (using an induction-based proof that $ k! \geq 2^{k-1} $ for $ k \geq 1 $).

Thus, $ \lim{} s_n = e \approx{} 2.71828 $.

Unbounded Monotone Sequences

[Theorem 10.4] If $ (s_n)_{n \geq{} m} $ is an unbounded increasing (decreasing) sequence, then $ \lim{} s_n = +\infty{} $ (resp. $ \lim{} s_n = -\infty{} $). [Harmonic Sequence] Let $ s_n = \sum_{k=1}^n \frac{1}{k} $. This sequence is:

1. Increasing, since $ s_{n+1} = s_n + \frac{1}{n+1} > s_n $.
2. Unbounded, as shown using a lower bound argument: $$\begin{align} s_{2^m} \geq \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{2^{m-1}} \geq m. \end{align}$$

Thus, $ s_n \to{} +\infty{} $.

Lim Sup and Lim Inf

[Lim Sup and Lim Inf] Let $ u_N = \sup{}\{s_n : n > N\} $ and $ v_N = \inf{}\{s_n : n > N\} $. Then: $$\begin{align} \limsup s_n = \lim_{N \to \infty} u_N, \quad \liminf s_n = \lim_{N \to \infty} v_N. \end{align}$$ [Properties of Lim Sup and Lim Inf]

1. $ \limsup s_n \geq \liminf s_n $.
2. If $ \lim s_n $ exists, then $ \limsup s_n = \lim s_n = \liminf s_n $.
3. If $ \limsup s_n = \liminf s_n = s $, then $ \lim s_n = s $.

[Oscillating Sequence] Let $$\begin{align} s_n = \begin{cases} \frac{1}{n}, & n \text{ even} \\ -n, & n \text{ odd} \end{cases}. \end{align}$$ Then:

1. $ \limsup s_n = 0 $,
2. $ \liminf s_n = -\infty $.

Decimal Expansions

Any real number can be expressed as a decimal expansion $ K.d_1d_2d_3\ldots{} $, where $ K \in{} \{0, 1, 2, \dots{} \} $ and $ d_k \in{} \{0, \ldots{}, 9\} $. For instance: $$\begin{align} 1 = 1.000\ldots = 0.999\ldots \end{align}$$

Lim Inf, Lim Sup, and Cauchy Sequences (Sections 10-11)

Lim Inf and Lim Sup

[Lim Sup and Lim Inf] Let $ u_N = \sup{}\{s_n : n > N\} $ and $ v_N = \inf{}\{s_n : n > N\} $. Then: $$\begin{align} \limsup s_n = \lim_{N \to \infty} u_N, \quad \liminf s_n = \lim_{N \to \infty} v_N. \end{align}$$ [Theorem 10.7]

1. If $ \lim s_n $ is defined, then $ \liminf s_n = \lim s_n = \limsup s_n $.
2. If $ \liminf s_n = s = \limsup s_n $, then $ \lim s_n = s $.

Let $$\begin{align} s_n = \begin{cases} \frac{1}{n} & \text{if } n \text{ is even}, \\ -n & \text{if } n \text{ is odd}. \end{cases} \end{align}$$ Then:

1. $ \limsup s_n = 0 $,
2. $ \liminf s_n = -\infty $.

Cauchy Sequences

[Cauchy Sequence] A sequence $ (s_n) $ is called Cauchy if: $$\begin{align} \forall \varepsilon > 0, \exists N \in \mathbb{N} \text{ such that } |s_n - s_m| < \varepsilon \text{ for } n, m > N. \end{align}$$ [Theorem 10.11] A sequence $ (s_n) $ converges if and only if it is Cauchy. [Lemma 10.9] Any convergent sequence is Cauchy. [Lemma 10.10] Any Cauchy sequence is bounded.

Subsequences

[Subsequence] A sequence $ (t_k) $ is a subsequence of $ (s_n) $ if there exists a strictly increasing sequence $ n_1 < n_2 < \ldots{} $ such that $ t_k = s_{n_k} $ for any $ k $. [Subsequence Examples]

1. $ s_n = \frac{1}{n} $: A subsequence $ t_k = \frac{1}{k^2} $, where $ n_k = k^2 $.
2. $ s_n = (-1)^n + \frac{1}{n} $: The sequence diverges, but $ s_{2k} = 1 + \frac{1}{2k} $ converges.

[Subsequential Limits] Every sequence has a subsequence that converges to a limit.

Subsequences and Subsequential Limits (Section 11)

Subsequences

[Subsequence] A sequence $ (t_k) $ is a subsequence of $ (s_n) $ if there exists a strictly increasing sequence $ n_1 < n_2 < \ldots{} $ such that $ t_k = s_{n_k} $ for any $ k $. [Subsequences of Subsequences] Any subsequence of a subsequence of $ (s_n) $ is a subsequence of $ (s_n) $.

Convergence of Subsequences

[Theorem 11.3] If $ \lim{} s_n = s $ (finite or $ \pm{}\infty{} $), then any subsequence $ (t_k) $ has the same limit.

Monotone Subsequences

[Theorem 11.4] Every sequence has a monotone subsequence. [Bolzano-Weierstrass Theorem] Every bounded sequence has a convergent subsequence. [Divergent Sequence with Convergent Subsequence] Let $ s_n = (-1)^n (1 + \frac{1}{n}) $. This sequence is bounded but divergent. The subsequence $ s_{2k} = 1 + \frac{1}{2k} $ converges to $ 1 $.

Subsequential Limits

[Subsequential Limit] A subsequential limit of $ (s_n) $ is any limit of a subsequence, possibly $ \pm{}\infty{} $. [Theorem 11.2] Suppose $ (s_n) $ is a sequence.

1. $ t \in \mathbb{R} $ is a subsequential limit if and only if $ \forall \varepsilon > 0, \{n : |s_n - t| < \varepsilon\} $ is infinite.
2. $ t = +\infty $ (or $ t = -\infty $) is a subsequential limit if $ (s_n) $ is not bounded above (or below).

Lim Inf and Lim Sup as Subsequential Limits

[Theorem 11.7] For any sequence $ (s_n) $, $ \limsup{} s_n $ and $ \liminf{} s_n $ are limits of monotone subsequences. [Theorem 11.8] Let $ S $ be the set of subsequential limits of $ (s_n) $. Then:

1. $ S $ is non-empty.
2. $ \inf S = \liminf s_n, \quad \sup S = \limsup s_n $.
3. $ \lim s_n $ exists if and only if $ S $ consists of a single point, $ S = \{\lim s_n\} $.

Lim Sup, Lim Inf, and Metric Spaces (Sections 11-13)

Subsequential Limits

[Subsequential Limit (Definition 11.6)] For a sequence $ (s_n) $, a subsequential limit is any limit of a subsequence (in $ \mathbb{R} \cup{} \{\pm{}\infty{}\} $). [Properties of Subsequential Limits (Theorem 11.2)] Suppose $ (s_n) $ is a sequence.

1. $ t \in \mathbb{R} $ is a subsequential limit if and only if $ \forall \varepsilon > 0, \{n : |s_n - t| < \varepsilon\} $ is infinite.
2. $ t = +\infty $ ($ t = -\infty $) is a subsequential limit if $ (s_n) $ is not bounded above (resp. below).

The Set of Subsequential Limits

[Theorem 11.8] Suppose $ (s_n) $ is a sequence, and $ S $ is the set of subsequential limits. Then:

1. $ S $ is non-empty.
2. $ \inf S = \liminf s_n $ and $ \sup S = \limsup s_n $.
3. $ \lim s_n $ exists if and only if $ S $ consists of a single point. Then $ \{\lim s_n\} = S $.

[Convergent Sequences] If $ \lim{} s_n = s $, then $ S = \{s\} $, $ \limsup{} s_n = s = \liminf{} s_n $.

Lim Sup and Lim Inf Revisited

[Theorem 12.1] If $ \lim{} s_n = s \in{} (0, \infty{}) $, then for any sequence $ (t_n) $: $$\begin{align} \limsup (s_n t_n) = s \cdot \limsup t_n. \end{align}$$ [Corollary 12.3] If $ (s_n) $ is a sequence of positive numbers and $ \lim{} \frac{s_{n+1}}{s_n} $ exists, then: $$\begin{align} \lim s_n^{1/n} \text{ also exists, and } \lim s_n^{1/n} = \lim \frac{s_{n+1}}{s_n}. \end{align}$$

1. $ \lim (n!)^{1/n} = +\infty $.
2. $ \lim \frac{1}{n} (n!)^{1/n} = \frac{1}{e} $.

Metric Spaces

[Metric (Definition 13.1)] A metric $ d : S \times{} S \to{} [0, \infty{}) $ satisfies:

1. {Non-degeneracy:} $ d(x, y) = 0 \iff x = y $.
2. {Symmetry:} $ d(x, y) = d(y, x) $ for all $ x, y \in S $.
3. {Triangle Inequality:} $ d(x, y) + d(y, z) \geq d(x, z) $ for all $ x, y, z \in S $.

[Metrics]

1. On $ \mathbb{R} $: $ d(x, y) = |x - y| $.
2. On $ \mathbb{R}^n $: $ d(\vec{x}, \vec{y}) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} $.

Convergence in Metric Spaces

[Convergence (Definition 13.2)] A sequence $ (s_n) \subset{} S $ converges to $ s \in{} S $ if: $$\begin{align} \lim_{n \to \infty} d(s_n, s) = 0. \end{align}$$ [Cauchy Sequence (Definition 13.2)] A sequence $ (s_n) \subset{} S $ is Cauchy if: $$\begin{align} \forall \varepsilon > 0, \exists N \text{ such that } d(s_n, s_m) < \varepsilon \text{ for all } n, m > N. \end{align}$$ [Cauchy and Convergence] If $ (s_n) $ converges in $ (S, d) $, then $ (s_n) $ is Cauchy. [Complete Metric Spaces] A metric space $ (S, d) $ is complete if every Cauchy sequence in $ S $ converges to a point in $ S $. The space $ \mathbb{R}^n $ with the Euclidean metric is complete.

Metric Spaces (Section 13)

Definition and Examples

[Metric (Definition 13.1)] Suppose $ S $ is a set. A function $ d : S \times{} S \to{} [0, \infty{}) $ is called a metric if the following hold:

1. {Non-degeneracy:} $ d(x, y) = 0 \iff x = y $ (hence $ d(x, y) > 0 $ when $ x \neq y $).
2. {Symmetry:} $ d(x, y) = d(y, x) $ for all $ x, y \in S $.
3. {Triangle Inequality:} $ d(x, y) + d(y, z) \geq d(x, z) $ for all $ x, y, z \in S $.

[Examples of Metrics]

1. On $ \mathbb{R} $: $ d(x, y) = |x - y| $.
2. On $ \mathbb{R}^n $: $ d(\vec{x}, \vec{y}) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} $ (Euclidean metric).
3. Discrete Metric: For $ x, y \in S $, define: $$\begin{align} d(x, y) = \begin{cases} 0 & \text{if } x = y, \\ 1 & \text{if } x \neq y. \end{cases} \end{align}$$

Convergence and Completeness

[Convergence (Definition 13.2)] Suppose $ (S, d) $ is a metric space. A sequence $ (s_n) \subset{} S $ converges to $ s \in{} S $ if $ \lim_{n \to \infty} d(s_n, s) = 0 $; that is, $ \forall{} \varepsilon{} > 0, \exists{} N \text{such that} d(s_n, s) < \varepsilon{} \text{for} n > N $. [Cauchy Sequence (Definition 13.2 continued)] A sequence $ (s_n) \subset{} S $ is called Cauchy if: $$\begin{align} \forall \varepsilon > 0, \exists N \text{ such that } d(s_n, s_m) < \varepsilon \text{ for } n, m > N. \end{align}$$ [Cauchy Sequences are Convergent in Complete Spaces] If $ (S, d) $ is complete, then every Cauchy sequence in $ S $ converges to a point in $ S $. [Completeness of $ \mathbb{R} $] $ \mathbb{R} $ with the standard metric $ d(x, y) = |x - y| $ is complete. Any Cauchy sequence in $ \mathbb{R} $ converges to a real number. [Non-Completeness of $ \mathbb{Q} $] Consider $ \mathbb{Q} $ with $ d(x, y) = |x - y| $. The sequence $ r_n \in{} (\sqrt{2} - \frac{1}{n}, \sqrt{2}) $ is Cauchy in $ \mathbb{Q} $ but does not converge in $ \mathbb{Q} $ because $ \sqrt{2} \notin{} \mathbb{Q} $.

Special Metrics

[Manhattan (Taxicab) Metric] For $ \vec{x} = (x_1, \ldots{}, x_n) $ and $ \vec{y} = (y_1, \ldots{}, y_n) $ in $ \mathbb{R}^n $, the taxicab metric is defined as: $$\begin{align} d_1(\vec{x}, \vec{y}) = \sum_{i=1}^n |x_i - y_i|. \end{align}$$ [Completeness of $ (\mathbb{R}^n, d_1) $] The metric space $ (\mathbb{R}^n, d_1) $ is complete.

Inner Product and Triangle Inequality

[Inner Product] For $ \vec{x}, \vec{y} \in{} \mathbb{R}^n $, define the inner product: $$\begin{align} \langle \vec{x}, \vec{y} \rangle = \sum_{i=1}^n x_i y_i. \end{align}$$ The magnitude of $ \vec{x} $ is: $$\begin{align} \|\vec{x}\| = \sqrt{\langle \vec{x}, \vec{x} \rangle}. \end{align}$$ [Bunyakovsky-Cauchy-Schwarz Inequality] For all $ \vec{x}, \vec{y} \in{} \mathbb{R}^n $: $$\begin{align} |\langle \vec{x}, \vec{y} \rangle| \leq \|\vec{x}\| \|\vec{y}\|. \end{align}$$ [Triangle Inequality Lite] For $ \vec{x}, \vec{y} \in{} \mathbb{R}^n $: $$\begin{align} \|\vec{x} + \vec{y}\| \leq \|\vec{x}\| + \|\vec{y}\|. \end{align}$$

Metric Spaces: Bounded Sets, Open and Closed Sets, and Closure (Section 13)

Bounded Sets

[Bounded Sets] A set $ E $ in a metric space $ (S, d) $ is bounded if there exists $ y \in{} S $ such that: $$\begin{align} \sup_{x \in E} d(y, x) < \infty. \end{align}$$ If such a $ y $ exists, then for any $ z \in{} S $, $ \sup_{x \in E} d(z, x) < \infty{} $. This follows from the triangle inequality: $$\begin{align} d(z, x) \leq d(y, x) + d(z, y). \end{align}$$ A sequence $ (x_k) $ is bounded if the set $ \{x_1, x_2, \ldots{}\} $ is bounded. That is, for some (or any) $ y \in{} S $: $$\begin{align} \sup_k d(y, x_k) < \infty. \end{align}$$

Bolzano-Weierstrass Theorem

[Bolzano-Weierstrass for $ \mathbb{R}^n $] Any bounded sequence in $ \mathbb{R}^n $ has a convergent subsequence. [Failure of Bolzano-Weierstrass in Discrete Metrics] Consider $ \mathbb{N} $ equipped with the discrete metric $ d(x, y) = 1 $ for $ x \neq{} y $ and $ d(x, y) = 0 $ for $ x = y $. The sequence $ x_n = n $ is bounded but has no convergent subsequences because convergent sequences are eventually constant in discrete metrics.

Interior Points and Open Sets

[Open Ball] An open ball with center $ s_0 $ and radius $ r > 0 $ is: $$\begin{align} B_r^o(s_0) = \{s \in S : d(s, s_0) < r\}. \end{align}$$ [Interior Points] A point $ s_0 \in{} S $ is interior to $ E \subset{} S $ if there exists $ r > 0 $ such that $ B_r^o(s_0) \subset{} E $. The set of all interior points is denoted $ E^o $, called the interior of $ E $. [Open Sets] A set $ E \subset{} S $ is open if $ E = E^o $.

1. In $ S = \mathbb{R} $ with the usual metric, $ [0, \infty) $ is not open, but $ (0, \infty) $ is.
2. In $ S = \mathbb{R}^2 $, the set $ E = \{(x, 0) : x \geq 0\} $ has $ E^o = \emptyset $.

Properties of Open and Closed Sets

[Facts about Open Sets]

1. $ S $ and $ \emptyset $ are open.
2. A union of any collection of open sets is open.
3. A finite intersection of open sets is open.

[Closed Sets] A set $ E \subset{} S $ is closed if $ S \setminus{} E $ is open. [Facts about Closed Sets]

1. $ S $ and $ \emptyset $ are closed.
2. An intersection of any collection of closed sets is closed.
3. A finite union of closed sets is closed.

Closure and Boundary

[Closure] The closure of $ E \subset{} S $, denoted $ \overline{E} $, is the intersection of all closed sets containing $ E $. [Boundary] The boundary of $ E \subset{} S $ is: $$\begin{align} \partial E = \overline{E} \setminus E^o. \end{align}$$ [Closure in $ \mathbb{R} $] Let $ E = \{\frac{1}{n} : n \in{} \mathbb{N}\} \subset{} \mathbb{R} $. Then: $$\begin{align} \overline{E} = E \cup \{0\}. \end{align}$$

Closure, Boundary, and Open/Closed Sets (Section 13)

Open Balls and Open Sets

[Open Ball] For $ s_0 \in{} S $ and $ r > 0 $, the open ball with center $ s_0 $ and radius $ r $ is: $$\begin{align} B_r^o(s_0) = \{s \in S : d(s, s_0) < r\}. \end{align}$$ A set $ E \subset{} S $ is open if and only if it is a union of open balls.

Closed Sets and De Morgan's Laws

[Closed Sets] A set $ E \subset{} S $ is closed if $ S \setminus{} E $ is open. [Properties of Closed Sets]

1. $ S $ and $ \emptyset $ are closed.
2. Any intersection of closed sets is closed.
3. A finite union of closed sets is closed.

[De Morgan's Laws] For any collection $ \{A_i\}_{i \in{} I} \subset{} S $: $$\begin{align} S \setminus \bigcup_{i \in I} A_i = \bigcap_{i \in I} (S \setminus A_i), \quad S \setminus \bigcap_{i \in I} A_i = \bigcup_{i \in I} (S \setminus A_i). \end{align}$$

Examples of Open and Closed Sets

[Intervals in $ \mathbb{R} $]

1. $ (a, b) $ is open, but not closed.
2. $ [a, b] $ is closed, but not open.
3. $ (a, b], [a, b) $ are neither open nor closed.

[Discrete Metric] In a discrete metric space:

1. Every set is both open and closed.

Closure and Boundary

[Closure] The closure of $ E \subset{} S $, denoted $ \overline{E} $, is the intersection of all closed sets containing $ E $. [Boundary] The boundary of $ E \subset{} S $ is: $$\begin{align} \partial E = \overline{E} \setminus E^o. \end{align}$$

Properties of Closure and Boundary

1. $ E = \overline{E} $ if and only if $ E $ is closed.
2. $ s \in \overline{E} $ if and only if $ s $ is a limit of a sequence in $ E $.
3. $ \partial E = \overline{E} \cap (S \setminus E)^- $.

[Closure of $ E = \{\frac{1}{n} : n \in{} \mathbb{N}\} \subset{} \mathbb{R} $] The closure is: $$\begin{align} \overline{E} = \left\{\frac{1}{n} : n \in \mathbb{N}\right\} \cup \{0\}. \end{align}$$

Compactness (Section 13)

Definition of Compactness

[Compactness (Definition 13.11)] Suppose $ E \subset{} S $. A family $ \mathcal{U} $ of open sets is an open cover for $ E $ if: $$\begin{align} E \subset \bigcup_{U \in \mathcal{U}} U. \end{align}$$ A subcover is a subfamily of $ \mathcal{U} $ which is also an open cover. $ E $ is called compact if every open cover has a finite subcover. A cover $ \mathcal{U} $ is a collection of sets, not their union. Thus, a cover is a subset of $ \mathcal{P}(S) $ (the power set of $ S $), not $ S $.

Examples in $ \mathbb{R

$ with Usual Metric}

1. $ E = [0, \infty) $ is not compact. For example:
   1. $ U_k = (-1, k) $ ($ k \in \mathbb{N} $) is an open cover with no finite subcover.
2. $ E = (0, 1) $ is not compact. For example:
   1. $ U_k = (1/k, 1) $ ($ k \in \mathbb{N} $) is an open cover with no finite subcover.
3. $ E = [a, b] $ ($ a, b \in \mathbb{R} $) is compact (proof to follow).

[Compactness of Finite Sets] Any finite set is compact.

Compactness and Boundedness

Any compact set is bounded.
[Bounded but Not Compact] Equip $ \mathbb{N} $ with the discrete metric $$\begin{align} d(x, y) = \begin{cases} 0 & x = y, \\ 1 & x \neq y. \end{cases} \end{align}$$ Then $ \mathbb{N} $ is bounded, but it is not compact because the open cover $ U_n = \{n\} $ ($ n \in{} \mathbb{N} $) has no finite subcover.

Properties of Compact Sets

1. A closed subset of a compact set is compact.
2. A finite union of compact sets is compact.

Nested Sequences of Closed Sets

Suppose $ F_1 \supset{} F_2 \supset{} \cdots{} $ are closed non-empty subsets of a compact set $ E $. Then: $$\begin{align} \bigcap_{n} F_n \neq \emptyset, \quad \text{and it is compact.} \end{align}$$

Heine-Borel Theorem and Cantor Set

[Heine-Borel Theorem] A subset of $ \mathbb{R}^n $ is compact if and only if it is closed and bounded. [Cantor Set] Define: $$\begin{align} F_0 = [0, 1], \quad F_1 = [0, 1/3] \cup [2/3, 1], \quad F_2 = [0, 1/9] \cup [2/9, 1/3] \cup [2/3, 7/9] \cup [8/9, 1], \dots \end{align}$$ The Cantor set $ C = \bigcap_n{} F_n $ is non-empty, closed, and compact. It contains no intervals, and its interior is empty.

Compactness and Total Boundedness (Section 13)

Definition of Compactness

[Compactness (Definition 13.11)] Suppose $ E \subset{} S $. A family $ \mathcal{U} $ of open sets is an open cover for $ E $ if: $$\begin{align} E \subset \bigcup_{U \in \mathcal{U}} U. \end{align}$$ A subcover is a subfamily of $ \mathcal{U} $ which is also an open cover. $ E $ is called compact if every open cover has a finite subcover. A cover $ \mathcal{U} $ is a collection of sets, not their union. Thus, a cover is a subset of $ \mathcal{P}(S) $ (the power set of $ S $), not $ S $.

Compactness and Completeness

Suppose $ E \subset{} S $. If $ E $ is compact, then $ E $ is complete.
Any compact set is closed. [Compactness Criterion] $ E \subset{} S $ is compact if and only if it is complete and totally bounded.

Total Boundedness

[Total Boundedness] A set $ E \subset{} S $ is called totally bounded if $ \forall{} \varepsilon{} > 0 $, there exist $ s_1, \ldots{}, s_n \in{} S $ such that: $$\begin{align} E \subset \bigcup_{i=1}^n B_\varepsilon^o(s_i). \end{align}$$ A set is totally bounded if and only if any sequence in the set has a Cauchy subsequence.

Characterization of Compactness

For a subset $ E $ of a metric space, the following are equivalent:

1. $ E $ is compact.
2. $ E $ is complete and totally bounded.
3. Any sequence in $ E $ has a subsequence with a limit in $ E $.

The space $ \mathbb{N} $ with the discrete metric is complete and bounded but not compact.

Heine-Borel Theorem

[Heine-Borel Theorem] A subset of $ \mathbb{R}^n $ is compact if and only if it is closed and bounded. [Cantor Set] The Cantor set $ C $, constructed as: $$\begin{align} F_0 = [0, 1], \quad F_1 = [0, 1/3] \cup [2/3, 1], \quad F_2 = \cdots, \end{align}$$ is compact, closed, and totally bounded but has no interior.

A Note on Compactness

Total Boundedness

[Total Boundedness (Definition 1.1)] A set $ S \subset{} E $ is called totally bounded if for every $ \varepsilon{} > 0 $, there exist $ p_1, \ldots{}, p_n \in{} E $ such that: $$\begin{align} S \subset \bigcup_{i=1}^n B_\varepsilon^o(p_i). \end{align}$$ [Intrinsic Nature of Total Boundedness (Proposition 1.2)] A set $ S \subset{} E $ is totally bounded if and only if for every $ \varepsilon{} > 0 $, there exist $ q_1, \ldots{}, q_m \in{} S $ such that: $$\begin{align} S \subset \bigcup_{j=1}^m B_\varepsilon^o(q_j). \end{align}$$ [Total Boundedness and Cauchy Subsequences (Proposition 1.3)] A set $ S $ is totally bounded if and only if any sequence in $ S $ has a Cauchy subsequence. [Characterization of Compactness (Corollary 1.4)] A set $ S $ is totally bounded and complete if and only if any sequence in $ S $ has a subsequence converging to a limit in $ S $.

Compactness

[Characterization of Compactness (Theorem 2.1)] For a subset $ S \subset{} E $, the following are equivalent:

1. $ S $ is compact.
2. Any sequence in $ S $ has a convergent subsequence.
3. $ S $ is complete and totally bounded.

[Heine-Borel Theorem (Theorem 3.1)] A set $ S \subset{} \mathbb{R}^n $ is compact if and only if it is closed and bounded. [Total Boundedness in $ \mathbb{R}^n $ (Lemma 3.3)] A set $ S \subset{} \mathbb{R}^n $ is bounded if and only if it is totally bounded. [Bolzano-Weierstrass Theorem (Theorem 3.4)] Every bounded sequence in $ \mathbb{R}^n $ has a convergent subsequence.

Compactness and Series (Sections 13-14)

Total Boundedness

[Total Boundedness] A set $ E \subset{} S $ is totally bounded if: $$\begin{align} \forall \varepsilon > 0, \exists s_1, \ldots, s_n \in S \text{ such that } E \subset \bigcup_{i=1}^n B_\varepsilon^o(s_i). \end{align}$$ A set $ E $ is totally bounded if and only if any sequence in $ E $ has a Cauchy subsequence.

Characterization of Compactness

For a subset $ E $ of a metric space, the following are equivalent:

1. $ E $ is compact.
2. $ E $ is complete and totally bounded.
3. Any sequence in $ E $ has a subsequence with a limit in $ E $.

[Heine-Borel] A subset of $ \mathbb{R}^n $ is compact if and only if it is closed and bounded. [Non-Compact Set] In $ \mathbb{N} $ with the discrete metric, $ \mathbb{N} $ is closed and bounded but not compact.

Series

[Series and Convergence] The $ n $-th partial sum of a series $ \sum_{j=k_0}^\infty{} a_j $ is: $$\begin{align} s_n = \sum_{j=k_0}^n a_j. \end{align}$$ The series converges if $ \lim_{n \to \infty} s_n $ exists, diverges otherwise. [Geometric Series] $$\begin{align} \sum_{j=0}^\infty r^j = \begin{cases} \frac{1}{1-r}, & |r| < 1, \\ \infty, & r \geq 1. \end{cases} \end{align}$$

Cauchy Criterion for Convergence

[Cauchy Criterion] A series $ \sum_j{} a_j $ satisfies the Cauchy criterion if: $$\begin{align} \forall \varepsilon > 0, \exists N \text{ such that } \left| \sum_{j=m}^n a_j \right| < \varepsilon \text{ for } n \geq m > N. \end{align}$$ A series converges if and only if it satisfies the Cauchy criterion.

Comparison Test for Convergence

[Comparison Test]

1. If $ |b_n| \leq a_n $ and $ \sum a_n $ converges, then $ \sum b_n $ converges.
2. If $ 0 \leq a_n \leq b_n $ and $ \sum b_n = \infty $, then $ \sum a_n = \infty $.

Series and Decimal Expansions (Sections 14 and 16)

Series

[Partial Sums and Convergence of Series] The $ n $-th partial sum of a series $ \sum_{j=k_0}^\infty{} a_j $ is: $$\begin{align} s_n = \sum_{j=k_0}^n a_j. \end{align}$$ The series $ \sum_{j=k_0}^\infty{} a_j $ converges if $ \lim_{n \to \infty} s_n $ exists. Otherwise, it diverges. [Geometric Series] $$\begin{align} \sum_{j=0}^\infty r^j = \begin{cases} \frac{1}{1-r}, & |r| < 1, \\ \text{diverges}, & r \geq 1 \text{ or } r \leq -1. \end{cases} \end{align}$$

Cauchy Criterion for Convergence

[Cauchy Criterion (Definition 14.3)] A series $ \sum_j{} a_j $ satisfies the Cauchy Criterion if: $$\begin{align} \forall \varepsilon > 0, \exists N \text{ such that } \left|\sum_{j=m}^n a_j\right| < \varepsilon \text{ for } n \geq m > N. \end{align}$$ [Cauchy Criterion (Theorem 14.4)] A series converges if and only if it satisfies the Cauchy Criterion.

Properties of Convergence

[Necessary Condition for Convergence (Corollary 14.5)] If $ \sum_j{} a_j $ converges, then $ \lim_{n \to \infty} a_n = 0 $. If $ a_n = \frac{1}{n} $, then $ \lim_{n \to \infty} a_n = 0 $, but $ \sum_{n=1}^\infty{} \frac{1}{n} $ diverges.

Tests for Convergence

[Comparison Test (Theorem 14.6)]

1. If $ 0 \leq |b_n| \leq a_n $ and $ \sum a_n $ converges, then $ \sum b_n $ converges.
2. If $ 0 \leq a_n \leq b_n $ and $ \sum b_n = \infty $, then $ \sum a_n = \infty $.

[Root Test (Theorem 14.9)] For a series $ \sum_n{} a_n $, let $ \alpha{} = \limsup_{n \to \infty} |a_n|^{1/n} $. Then:

1. The series converges absolutely if $ \alpha < 1 $.
2. The series diverges if $ \alpha > 1 $.
3. If $ \alpha = 1 $, the test gives no information.

[Ratio Test (Theorem 14.8)] For a series $ \sum_n{} a_n $ of nonzero terms:

1. The series converges absolutely if $ \limsup_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1 $.
2. The series diverges if $ \liminf_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| > 1 $.
3. If $ \liminf_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| \leq 1 \leq \limsup_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| $, the test gives no information.

Decimal Expansions

[Decimal Expansions (Theorem 16.2)] Any real number $ x \geq{} 0 $ has at least one decimal expansion: $$\begin{align} x = K.d_1d_2d_3\ldots = K + \sum_{j=1}^\infty \frac{d_j}{10^j}, \end{align}$$ where $ K \in{} \mathbb{Z} $ and $ d_j \in{} \{0, 1, \ldots{}, 9\} $. [Uniqueness of Decimal Expansions (Theorem 16.3)] Any $ x \geq{} 0 $ has either exactly one decimal expansion or exactly two, one ending in $ \ldots{} d000\ldots{} $ and the other in $ \ldots{} [d-1]999\ldots{} $. [Repeating Decimals (Theorem 16.5)] A real number $ x $ is rational if and only if its decimal expansion is repeating.

Series and Decimal Expansions (Sections 17 and 21)

Root and Ratio Tests for Series Convergence

[Root Test] For a series $ \sum_{n} a_n $, let $ \alpha{} = \limsup_{n \to \infty} |a_n|^{1/n} $. Then:

1. The series converges absolutely if $ \alpha < 1 $.
2. The series diverges if $ \alpha > 1 $.
3. If $ \alpha = 1 $, the test gives no information.

[Ratio Test (Theorem 14.8)] For a series $ \sum_{n} a_n $ of nonzero terms:

1. The series converges absolutely if $ \limsup_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 $.
2. The series diverges if $ \liminf_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1 $.
3. If $ \liminf_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \leq 1 \leq \limsup_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $, the test gives no information.

1. Consider $ \sum_{k=1}^\infty \frac{k^4}{2^k} $. Using the Root Test: $$\begin{align} a_k^{1/k} = \left( k^{1/k} \right)^4 \frac{1}{2}, \quad \lim_{k \to \infty} a_k^{1/k} = \frac{1}{2} < 1, \end{align}$$ so the series converges.
2. The $ p $-series $ \sum_{k=1}^\infty \frac{1}{k^p} $ converges if and only if $ p > 1 $. The Root and Ratio Tests are inconclusive for this series.

Decimal Expansions

[Decimal Expansion] For $ x \in{} [0, \infty{}) $, the decimal expansion of $ x $ is: $$\begin{align} x = K .d_1 d_2 d_3 \ldots = K + \sum_{j=1}^\infty \frac{d_j}{10^j}, \end{align}$$ where $ K \in{} \{0, 1, 2, \ldots{}\} $ and $ d_1, d_2, \ldots{} \in{} \{0, 1, \ldots{}, 9\} $. [Existence of Decimal Expansions (Theorem 16.2)] Any real number $ x \geq{} 0 $ has at least one decimal expansion. [Uniqueness of Decimal Expansions (Theorem 16.3)] Any $ x \geq{} 0 $ has either exactly one decimal expansion or exactly two:

1. One ending in $ \ldots d000 \ldots $, where $ d \in \{1, \ldots, 9\} $,
2. Another ending in $ \ldots (d-1)999 \ldots $.

For example, $ \frac{1}{2} = 0.5000\ldots{} = 0.4999\ldots{} $.

Repeating Decimal Expansions

[Repeating Decimals (Definition 16.4)] A repeating decimal expansion is one of the form: $$\begin{align} K .d_1 \ldots d_\ell d_{\ell+1} \ldots d_{\ell+r} = K .d_1 \ldots d_\ell \overline{d_{\ell+1} \ldots d_{\ell+r}}, \end{align}$$ where the sequence $ d_{\ell{}+1} \ldots{} d_{\ell{}+r} $ repeats. [Repeating Decimals and Rational Numbers (Theorem 16.5)] A real number $ x $ is rational if and only if its decimal expansion is repeating.

Continuity in Metric Spaces (Sections 17 and 21)

Definition of Continuity

[Continuity (Definition 21.1)] Suppose $ (S, d) $ and $ (S^*, d^*) $ are metric spaces. The function $ f : \text{dom}(f) \to{} S^* $ (with $ \text{dom}(f) \subset{} S $) is continuous at $ x \in{} \text{dom}(f) $ if: $$\begin{align} \forall \varepsilon > 0, \exists \delta > 0 \text{ such that } d^*(f(x), f(y)) < \varepsilon \text{ whenever } d(x, y) < \delta. \end{align}$$ $ f $ is called continuous on $ E \subset{} S $ if it is continuous at every $ x \in{} E $. [Sequential Criterion for Continuity (Theorem 17.1 + 17.2)] $ f : S \to{} S^* $ is continuous at $ x \in{} S $ if and only if $ f(x_n) \to{} f(x) $ whenever $ x_n \to{} x $.

Examples of Continuity and Discontinuity

[Discontinuous Everywhere] The Dirichlet function $ f : \mathbb{R} \to{} \mathbb{R} $, defined as: $$\begin{align} f(x) = \begin{cases} 1 & x \in \mathbb{Q}, \\ 0 & x \notin \mathbb{Q}, \end{cases} \end{align}$$ is discontinuous at every $ x \in{} \mathbb{R} $. This is because for any $ x \in{} \mathbb{R} $, one can find sequences $ (x_n) \subset{} \mathbb{Q} $ and $ (y_n) \not{}\subset{} \mathbb{Q} $ such that $ x_n, y_n \to{} x $, but $ f(x_n) \to{} 1 $ and $ f(y_n) \to{} 0 $, which do not match $ f(x) $. [Continuous at a Single Point] The modified Dirichlet function $ g : \mathbb{R} \to{} \mathbb{R} $, defined as: $$\begin{align} g(x) = \begin{cases} x & x \in \mathbb{Q}, \\ 0 & x \notin \mathbb{Q}, \end{cases} \end{align}$$ is continuous only at $ x = 0 $. At other points, similar reasoning as the Dirichlet function applies. [Continuous on $ \mathbb{R} \setminus{} \mathbb{Q} $] The Thomae function $ h : \mathbb{R} \to{} \mathbb{R} $, defined as: $$\begin{align} h(x) = \begin{cases} \frac{1}{b} & x = \frac{a}{b}, \, \gcd(a, b) = 1, \, b > 0, \, x \neq 0, \\ 0 & x \notin \mathbb{Q}, \end{cases} \end{align}$$ is continuous at $ x \notin{} \mathbb{Q} $ and discontinuous at $ x \in{} \mathbb{Q} $.

Operations on Continuous Functions

Suppose $ f, g $ are continuous at $ x_0 $ in a metric space $ (S, d) $. Then the following functions are also continuous at $ x_0 $:

1. $ |f| $,
2. $ kf $ ($ k \in \mathbb{R} $),
3. $ f + g $,
4. $ f \cdot g $,
5. $ f / g $ (if $ g(x_0) \neq 0 $).

If $ f, g $ are continuous at $ x_0 $, then $ \max{}(f, g) $ and $ \min{}(f, g) $ are continuous at $ x_0 $.

Composition of Continuous Functions

Suppose $ (S_1, d_1), (S_2, d_2), (S_3, d_3) $ are metric spaces, and $ f : \text{dom}(f) \to{} S_2 $, $ g : \text{dom}(g) \to{} S_3 $ are functions such that $ f $ is continuous at $ x_0 $, $ g $ is continuous at $ f(x_0) $, and $ x_0 \in{} \text{dom}(f) $. Then $ g \circ{} f $ is continuous at $ x_0 $.

Characterization of Continuity

[Characterization of Continuity (Theorem 21.3)] Suppose $ (S, d) $ and $ (S^*, d^*) $ are metric spaces. $ f : S \to{} S^* $ is continuous if and only if $ f^{-1}(U) $ is open for every open $ U \subset{} S^* $, where: $$\begin{align} f^{-1}(U) = \{s \in S : f(s) \in U\}. \end{align}$$ $ f $ is continuous at $ s_0 \in{} S $ if for any open set $ U $ containing $ f(s_0) $, there exists an open set $ V $ containing $ s_0 $ such that $ f(V) \subset{} U $.

Continuity and the Intermediate Value Theorem (Sections 18 and 21)

Another Characterization of Continuity

[Theorem 21.3] Suppose $ (S, d) $ and $ (S^*, d^*) $ are metric spaces. A function $ f : S \to{} S^* $ is continuous if and only if $ f^{-1}(U) $ is open for every open $ U \subset{} S^* $. Here: $$\begin{align} f^{-1}(U) = \{s \in S : f(s) \in U\}. \end{align}$$ [Exercise 21.2] $ f $ is continuous at $ s_0 \in{} S $ if and only if for any open set $ U \ni{} f(s_0) $, there exists an open set $ V \ni{} s_0 $ such that $ f(V) \subset{} U $. [Exercise 21.4] Suppose $ (S, d) $ is a metric space. A function $ f : S \to{} \mathbb{R} $ is continuous if and only if $ f^{-1}((a, b)) $ is open whenever $ a < b $.

Continuous Image of a Compact Set

If $ f : S \to{} S^* $ is continuous, and $ E \subset{} S $ is compact, then $ f(E) \subset{} S^* $ is compact. If $ f : S \to{} \mathbb{R} $ is continuous, and $ E \subset{} S $ is compact, then $ f(E) $ is bounded. Moreover, $ f $ attains its maximum and minimum values, i.e., there exist $ x, y \in{} E $ such that: $$\begin{align} f(x) = \sup_{e \in E} f(e), \quad f(y) = \inf_{e \in E} f(e). \end{align}$$

Intermediate Value Theorem (IVT)

[Theorem 18.2] Suppose $ I \subset{} \mathbb{R} $ is an interval, and $ f : I \to{} \mathbb{R} $ is continuous. Then $ f $ has the Intermediate Value Property (IVP) on $ I $: if $ a, b \in{} I $ with $ a < b $, and $ y $ lies between $ f(a) $ and $ f(b) $, then there exists $ x \in{} (a, b) $ such that $ f(x) = y $. If $ I $ is an interval, and $ f : I \to{} \mathbb{R} $ has the IVP, then $ f(I) $ is either an interval or a single point.

Applications of IVT

[Roots of Polynomials] Any polynomial of odd degree has at least one real root. [Existence of Fixed Points] Any continuous function $ f : [0, 1] \to{} [0, 1] $ has a fixed point, i.e., $ x \in{} [0, 1] $ such that $ f(x) = x $. [Existence of $ m $-th Roots] For any $ m \in{} \mathbb{N} $ and $ y > 0 $, there exists $ x > 0 $ such that $ x^m = y $.

Continuity of Inverse Functions

[Theorem 18.4] Suppose $ I \subset{} \mathbb{R} $ is an interval, and $ f : I \to{} \mathbb{R} $ is strictly increasing and continuous. Then $ J = f(I) $ is an interval, and $ f^{-1} : J \to{} I $ is strictly increasing and continuous. The function $ x \mapsto{} x^{1/m} $, taking $ [0, \infty{}) $ to itself, is continuous.

Continuity and Compactness (Section 18)

Continuous Image of a Compact Set

[Theorem 21.4(i)] Suppose $ f : S \to{} S^* $ is continuous, where $ (S, d) $ and $ (S^*, d^*) $ are metric spaces, and $ E \subset{} S $ is compact. Then $ f(E) \subset{} S^* $ is compact.

Maximum and Minimum of Continuous Functions

[Similar to 18.1] If $ f : S \to{} \mathbb{R} $ is continuous and $ E \subset{} S $ is compact, then $ f(E) $ is bounded. Moreover, $ f $ attains its maximum and minimum values, i.e., there exist $ x, y \in{} E $ such that: $$\begin{align} f(x) = \sup_{e \in E} f(e), \quad f(y) = \inf_{e \in E} f(e). \end{align}$$

Intermediate Value Property

[IVP] Suppose $ I \subset{} \mathbb{R} $ is an interval, and $ f : I \to{} \mathbb{R} $ is a function. $ f $ has the Intermediate Value Property (IVP) on $ I $ if for any $ a, b \in{} I $ with $ a < b $, and any $ y $ between $ f(a) $ and $ f(b) $, there exists $ x \in{} (a, b) $ such that $ f(x) = y $. [Theorem 18.2] Any continuous function has the IVP.

Applications of IVP

[18.3] If $ I $ is an interval, and $ f : I \to{} \mathbb{R} $ has the IVP, then $ f(I) $ is either an interval or a single point. [Roots of Polynomials] Any polynomial of odd degree has at least one real root. [Existence of Fixed Points] Any continuous function $ f : [0, 1] \to{} [0, 1] $ has a fixed point, i.e., a point $ x \in{} [0, 1] $ such that $ f(x) = x $. [Existence of $ m $-th Root] For any $ m \in{} \mathbb{N} $ and $ y > 0 $, there exists $ x > 0 $ such that $ x^m = y $.

Continuity of Inverse Functions

[Theorem 18.4] Suppose $ I \subset{} \mathbb{R} $ is an interval, and $ f : I \to{} \mathbb{R} $ is strictly increasing and continuous. Then $ f(I) $ is an interval, and $ f^{-1} : f(I) \to{} I $ is strictly increasing and continuous. The function $ x \mapsto{} x^{1/m} $, taking $ [0, \infty{}) $ to itself, is continuous.

Monotonicity of Injective Functions

[Theorem 18.6] Suppose $ f : I \to{} \mathbb{R} $ is a continuous one-to-one function on an interval $ I $. Then $ f $ is strictly monotone.

Uniform Continuity and Lipschitz Functions (Sections 18-19)

Uniform Continuity

[Uniform Continuity (Definition 21.1)] Suppose $ (S, d) $ and $ (S^*, d^*) $ are metric spaces. A function $ f : S \to{} S^* $ is uniformly continuous on $ E \subset{} S $ if: $$\begin{align} \forall \varepsilon > 0, \exists \delta > 0 \text{ such that } d^*(f(x), f(y)) < \varepsilon \text{ whenever } d(x, y) < \delta. \end{align}$$ Here, $ \delta{} $ depends only on $ \varepsilon{} $ and not on the specific point $ x $. [Sequential Criterion for Uniform Continuity (Theorem 19.4)] If $ f : S \to{} S^* $ is uniformly continuous, then $ f $ maps Cauchy sequences in $ S $ to Cauchy sequences in $ S^* $. [Non-Uniformly Continuous Function] The function $ f(x) = \frac{1}{x} $ is not uniformly continuous on $ (0, \infty{}) $, as the Cauchy sequence $ x_n = \frac{1}{n} $ is mapped to $ f(x_n) = n $, which is not Cauchy.

Lipschitz Functions

[Lipschitz Continuity] A function $ f : S \to{} S^* $ is Lipschitz if there exists $ K > 0 $ (called the Lipschitz constant) such that: $$\begin{align} d^*(f(s), f(t)) \leq K \cdot d(s, t), \quad \forall s, t \in S. \end{align}$$ Any Lipschitz function is uniformly continuous.
For $ a > 0 $, $ f(x) = \frac{1}{x} $ is Lipschitz (hence uniformly continuous) on $ [a, \infty{}) $. [Uniformly Continuous but Not Lipschitz] The function $ f(x) = \sqrt{x} $ is uniformly continuous on $ [0, \infty{}) $ but not Lipschitz.

Uniform Continuity on Compact Sets

[Uniform Continuity on Compact Sets (Theorem 21.4(ii))] If $ f : S \to{} S^* $ is continuous, and $ E \subset{} S $ is compact, then $ f $ is uniformly continuous on $ E $.

Uniform Continuity and Connectedness (Section 22)

Uniform Continuity on Compact Sets

[Theorem 21.4(ii)] Suppose $ (S, d) $ and $ (S^*, d^*) $ are metric spaces, and $ f : S \to{} S^* $ is continuous. If $ E \subset{} S $ is compact, then $ f|_E $ is uniformly continuous.

Extension of Uniformly Continuous Functions

Suppose $ E \subset{} S $ is compact, $ f : E \to{} S^* $, and $ S^* $ is complete. Then $ f $ is uniformly continuous if and only if it extends to a continuous $ \tilde{f} : \overline{E} \to{} S^* $.

Connectedness

[Connected and Disconnected Sets] Suppose $ (S, d) $ is a metric space. A set $ E \subset{} S $ is disconnected if there exist open sets $ U_1, U_2 \subset{} S $ such that:

1. $ E \subset U_1 \cup U_2 $,
2. $ (E \cap U_1) \cap (E \cap U_2) = \emptyset $,
3. $ E \cap U_1 \neq \emptyset $ and $ E \cap U_2 \neq \emptyset $.

A set $ E $ is connected if it is not disconnected. $ E $ is disconnected if and only if there exist $ A, B \subset{} E $ such that: $$\begin{align} E = A \cup B, \quad A \neq \emptyset, \quad B \neq \emptyset, \quad A \cap \overline{B} = \emptyset, \quad \overline{A} \cap B = \emptyset. \end{align}$$

Connectedness of Intervals

Any interval $ I \subset{} \mathbb{R} $ is connected.

Connectedness and Path Connectedness (Section 22)

Connectedness

[Connected Set (Definition 22.1)] Suppose $ (S, d) $ is a metric space. A set $ E \subset{} S $ is called disconnected if there exist open sets $ U_1, U_2 \subset{} S $ such that:

1. $ E \subset U_1 \cup U_2 $,
2. $ (E \cap U_1) \cap (E \cap U_2) = \emptyset $,
3. $ E \cap U_1 \neq \emptyset $ and $ E \cap U_2 \neq \emptyset $.

A set $ E $ is connected if it is not disconnected. An open set $ E $ is disconnected if and only if $ E = E_1 \cup{} E_2 $, where $ E_1, E_2 $ are disjoint, non-empty, open subsets. [Equivalent Characterization of Connectedness] A set $ E $ is disconnected if and only if there exist $ A, B \subset{} E $ such that: $$\begin{align} E = A \cup B, \quad A \neq \emptyset, \quad B \neq \emptyset, \quad A \cap \overline{B} = \emptyset, \quad \overline{A} \cap B = \emptyset. \end{align}$$

Continuous Images of Connected Sets

[Theorem 22.2] Suppose $ (S, d) $ and $ (S^*, d^*) $ are metric spaces. If $ E \subset{} S $ is connected and $ f : S \to{} S^* $ is continuous, then $ f(E) $ is connected.

Path Connectedness

[Path Connectedness (Definition 22.4)] A set $ E \subset{} S $ is path connected if for all $ a, b \in{} E $, there exists a continuous function $ \gamma{} : [0, 1] \to{} E $ such that $ \gamma{}(0) = a $ and $ \gamma{}(1) = b $. [Path Connected Sets Are Connected (Theorem 22.5)] Every path connected set is connected.

Connected but Not Path Connected Sets

Consider $ E \subset{} \mathbb{R}^2 $, where: $$\begin{align} E_1 = \{(0, y) : y \in (0, 1]\}, \quad E_2 = \{(x, 0) : x \in (0, 1]\} \cup \bigcup_{n \in \mathbb{N}} \{(1/n, y) : y \in (0, 1]\}. \end{align}$$ Then $ E = E_1 \cup{} E_2 $ is connected but not path connected.

Convex Sets

[Convex Sets] A set $ E \subset{} \mathbb{R}^n $ is convex if for all $ \vec{x}, \vec{y} \in{} E $ and $ t \in{} [0, 1] $, the point: $$\begin{align} \vec{z} = (1-t)\vec{x} + t\vec{y} \in E. \end{align}$$ Any convex set is path connected.

Path Connectedness of Graphs

The graph of a function $ f : I \to{} \mathbb{R} $, where $ I \subset{} \mathbb{R} $ is an interval, is path connected if and only if $ f $ is continuous.

Graphs of Functions and Path Connectedness (Sections 22-23-24)

Graphs and Path Connectedness

[Graph of a Function] The graph of a function $ f : I \to{} \mathbb{R} $ (where $ I \subset{} \mathbb{R} $ is an interval) is: $$\begin{align} G(f) = \{(x, f(x)) : x \in I\}. \end{align}$$ [Example 4 from Section 22] $ G(f) $ is path connected if and only if $ f $ is continuous on $ I $. [Discontinuous $ f $ with Connected $ G(f) $] Exercise 22.4 describes a function $ f $ such that $ G(f) $ is connected but $ f $ is discontinuous. [Multivariate Continuity] The function $ f : S \to{} \mathbb{R}^n $, $ x \mapsto{} (f_1(x), \ldots{}, f_n(x)) $, is continuous if and only if each $ f_i : S \to{} \mathbb{R} $ is continuous for $ 1 \leq{} i \leq{} n $.

Power Series

[Power Series] A power series is a series of the form: $$\begin{align} \sum_{n=0}^\infty a_n x^n, \end{align}$$ where $ x $ is a variable. [Radius of Convergence] Let $ \beta{} = \limsup{} |a_n|^{1/n} $ and $ R = 1/\beta{} $. The series: $$\begin{align} \sum_{n=0}^\infty a_n x^n \end{align}$$ converges for $ |x| < R $, diverges for $ |x| > R $. $ R $ is called the radius of convergence. If $ \lim{} |a_{n+1}/a_n| $ exists, it equals $ \beta{} $. The series may converge or diverge at $ \pm{} R $. The interval of convergence is one of: $$\begin{align} (-R, R), \, [-R, R), \, (-R, R], \, \text{or} \, [-R, R]. \end{align}$$

Examples of Power Series

1. $ \sum_{n=0}^\infty \frac{x^n}{n!} $: $ a_n = \frac{1}{n!} $, $ \beta = 0 $, $ R = \infty $. Interval: $ (-\infty, \infty) $. $$\begin{align} \sum_{n=0}^\infty \frac{x^n}{n!} = e^x. \end{align}$$
2. $ \sum_{n=0}^\infty x^n $: $ a_n = 1 $, $ \beta = 1 $, $ R = 1 $. Diverges for $ x = \pm 1 $. Interval: $ (-1, 1) $. $$\begin{align} \sum_{n=0}^\infty x^n = \frac{1}{1-x}. \end{align}$$
3. $ \sum_{n=0}^\infty \frac{x^n}{n+1} $: $ a_n = \frac{1}{n+1} $, $ \beta = 1 $, $ R = 1 $. Diverges at $ x = 1 $, converges for $ x \in [-1, 1) $. $$\begin{align} \sum_{n=0}^\infty \frac{x^n}{n+1} = \ln(1-x). \end{align}$$
4. $ \sum_{n=0}^\infty \frac{x^n}{(n+1)^2} $: $ a_n = \frac{1}{(n+1)^2} $, $ \beta = 1 $, $ R = 1 $. Converges for $ x \in [-1, 1] $.
5. $ \sum_{n=0}^\infty n! x^n $: $ a_n = n! $, $ \beta = \infty $, $ R = 0 $. Diverges for all $ x \neq 0 $.

Uniform Convergence

[Uniform Convergence (Definition 24.1-2)] A sequence $ f_n \to{} f $ pointwise on $ S $ if: $$\begin{align} \forall x \in S, \, \forall \varepsilon > 0, \exists N \text{ such that } |f_n(x) - f(x)| < \varepsilon \text{ for } n \geq N. \end{align}$$ It converges uniformly if: $$\begin{align} \forall \varepsilon > 0, \exists N \text{ such that } \sup_{x \in S} |f_n(x) - f(x)| < \varepsilon \text{ for } n \geq N. \end{align}$$ [Preservation of Continuity (Theorem 24.3)] If $ f_n \to{} f $ uniformly on $ S $ and each $ f_n $ is continuous, then $ f $ is continuous. Consider $ f_n(x) = n^2 x^n (1-x) $ on $ [0, 1] $. It converges pointwise to $ f(x) = 0 $, but not uniformly.

Uniform Convergence and Series of Functions (Sections 24-25)

Uniform Convergence

[Pointwise and Uniform Convergence (24.1-2)] Suppose $ f, f_1, f_2, \ldots{} $ are functions $ S \to{} \mathbb{R} $.

1. $ f_n \to f $ pointwise on $ S $ if: $$\begin{align} \forall x \in S, \, \forall \varepsilon > 0, \, \exists N \in \mathbb{N} \text{ such that } |f_n(x) - f(x)| < \varepsilon \text{ for } n \geq N. \end{align}$$
2. $ f_n \to f $ uniformly on $ S $ if: $$\begin{align} \forall \varepsilon > 0, \, \exists N \in \mathbb{N} \text{ such that } |f_n(x) - f(x)| < \varepsilon \text{ for } n \geq N, \, \forall x \in S. \end{align}$$ Equivalently, $ \lim_{n \to \infty} \sup_{x \in S} |f_n(x) - f(x)| = 0 $.

[Preservation of Continuity (24.3)] If $ f_n \to{} f $ uniformly on $ S $, and each $ f_n $ is continuous at $ x_0 \in{} S $, then $ f $ is continuous at $ x_0 $.

Uniformly Cauchy Sequences

[Uniformly Cauchy (25.3)] A sequence $ (f_n) $ of functions $ S \to{} \mathbb{R} $ is uniformly Cauchy if: $$\begin{align} \forall \varepsilon > 0, \, \exists N \in \mathbb{N} \text{ such that } |f_i(x) - f_j(x)| < \varepsilon \, \forall x \in S \text{ for } i, j \geq N. \end{align}$$ Equivalently, $ \sup_{x \in S} |f_i(x) - f_j(x)| < \varepsilon{} $ for $ i, j \geq{} N $. [Uniformly Cauchy $ \iff{} $ Uniform Convergence (25.4)] A sequence $ (f_n) $ is uniformly Cauchy if and only if it converges uniformly to some $ f $.

Series of Functions

[Convergence of Series] A series $ \sum_{n=1}^\infty{} g_n(x) $ converges (uniformly) if the sequence of partial sums $ s_k(x) = \sum_{n=1}^k g_n(x) $ converges (uniformly). [Uniform Convergence Preserves Continuity (25.5)] If $ g_n : S \to{} \mathbb{R} $ are continuous and $ \sum_{n=1}^\infty{} g_n(x) $ converges uniformly on $ S $, then $ \sum_{n=1}^\infty{} g_n(x) $ is continuous.

Weierstrass $ M $-Test

[Weierstrass $ M $-Test (25.7)] Suppose $ M_1, M_2, \ldots{} \geq{} 0 $ and $ \sum_{k=1}^\infty{} M_k < \infty{} $. If $ |g_k(x)| \leq{} M_k $ for all $ x \in{} S $ and $ k $, then $ \sum_{k=1}^\infty{} g_k(x) $ converges uniformly on $ S $. A power series $ \sum_{k=0}^\infty{} a_k x^k $ converges uniformly (to a continuous function) on $ [-b, b] $ if $ b < R $, where $ R = (\limsup{} |a_k|^{1/k})^{-1} $. Convergence need not be uniform on $ (-R, R) $. For example, $ \sum_{k=0}^\infty{} x^k = \frac{1}{1-x} $ converges on $ (-1, 1) $, but not uniformly because the partial sums are bounded while $ \frac{1}{1-x} $ is not.

Limits and Differentiation (Sections 20, 28-29)

Limits

[Limit (20.1, slightly modified)] Suppose $ S \subset{} \mathbb{R} $, $ a \in{} S^- $, $ f : S \to{} \mathbb{R} $, and $ L \in{} \mathbb{R} \cup{} \{\pm{} \infty{}\} $. Then: $$\begin{align} \lim_{x \to a, S} f = L \end{align}$$ if $ \lim{} f(x_n) = L $ for any sequence $ (x_n) \subset{} S $ with $ \lim{} x_n = a $. Such sequences $ (x_n) $ exist because $ a \in{} S^- $. [Connection Between Limits and Continuity] If $ a \in{} S $, then $ f : S \to{} \mathbb{R} $ is continuous at $ a $ if and only if $ \lim_{x \to a, S} f = f(a) $.

Common Set-Ups for Limits

1. **Usual Limit**: Let $ I $ be an interval, $ a $ be interior to $ I $, and $ S = I \setminus \{a\} $. Write $ \lim_{x \to a} f $ instead of $ \lim_{x \to a, S} f $.
2. **One-Sided Limit**: For $ S = (a, b) $, write $ \lim_{x \to a^+} f $ (right-hand limit). Define $ \lim_{x \to a^-} f $ similarly.

Useful Theorems About Limits

[Equivalent Definition of Limits (20.6, Simplified)] Suppose $ a \in{} S^- $. For $ f : S \to{} \mathbb{R} $ and $ L \in{} \mathbb{R} $, the following are equivalent:

1. $ \lim_{x \to a, S} f = L $.
2. For all $ \varepsilon > 0 $, there exists $ \delta > 0 $ such that $ |f(x) - L| < \varepsilon $ whenever $ x \in (a - \delta, a + \delta) \cap S \setminus \{a\} $.

[Limit Operations (20.4)] Suppose $ \lim_{x \to a, S} f_1 = L_1 $ and $ \lim_{x \to a, S} f_2 = L_2 $. Then:

1. $ \lim_{x \to a, S} (f_1 + f_2) = L_1 + L_2 $,
2. $ \lim_{x \to a, S} (f_1 \cdot f_2) = L_1 \cdot L_2 $,
3. If $ L_2 \neq 0 $, $ \lim_{x \to a, S} \frac{f_1}{f_2} = \frac{L_1}{L_2} $.

[Squeeze Theorem] If $ f(x) \leq{} g(x) \leq{} h(x) $ for all $ x \in{} S $, and $ \lim_{x \to a, S} f = \lim_{x \to a, S} h = L $, then $ \lim_{x \to a, S} g = L $.

Differentiation

[Derivative (28.1)] Suppose $ I $ is an open interval, $ a \in{} I $, and $ f : I \to{} \mathbb{R} $. The derivative of $ f $ at $ a $ is: $$\begin{align} f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}, \end{align}$$ if the limit exists and is finite.

Rules of Differentiation

[Product Rule] If $ f $ and $ g $ are differentiable at $ a $, then $ (fg)'(a) = f'(a)g(a) + f(a)g'(a) $. [Chain Rule (28.4)] If $ f $ is differentiable at $ a $, and $ g $ is differentiable at $ f(a) $, then $ g \circ{} f $ is differentiable at $ a $, with: $$\begin{align} (g \circ f)'(a) = g'(f(a)) \cdot f'(a). \end{align}$$

Carathéodory’s Theorem

[Carathéodory (Exercise 28.16)] Suppose $ I $ is an interval, $ f : I \to{} \mathbb{R} $. $ f $ is differentiable at $ a \in{} I $ if and only if there exists a function $ \phi{} : I \to{} \mathbb{R} $, continuous at $ a $, such that: $$\begin{align} f(x) - f(a) = \phi(x) \cdot (x - a), \quad \forall x \in I, \end{align}$$ and $ \phi{}(a) = f'(a) $.

Rules of Differentiation and Mean Value Theorem (Sections 28-29)

Definition of Derivative

[Derivative (28.1)] Suppose $ I $ is an open interval and $ a \in{} I $. A function $ f : I \to{} \mathbb{R} $ is differentiable at $ a $ if the derivative: $$\begin{align} f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \end{align}$$ exists and is finite.

Rules of Differentiation

[Product Rule (28.3)] Suppose $ f $ and $ g $ are differentiable at $ a $. Then $ fg $ is differentiable at $ a $, with: $$\begin{align} (fg)'(a) = f'(a)g(a) + f(a)g'(a). \end{align}$$ If $ f(x) = x^m $ for $ m \in{} \mathbb{N} $, then: $$\begin{align} (x^m)' = mx^{m-1}. \end{align}$$ [Chain Rule (28.4)] Suppose $ f $ is differentiable at $ a $ and $ g $ is differentiable at $ f(a) $. Then $ g \circ{} f $ is differentiable at $ a $, with: $$\begin{align} (g \circ f)'(a) = g'(f(a))f'(a). \end{align}$$

Examples of Differentiation

1. If $ f(x) = x^n $ for $ n \in \mathbb{N} $, then $ f'(x) = nx^{n-1} $.
2. If $ f(x) = x^{-n} $ for $ n \in \mathbb{N} $, then $ f'(x) = -nx^{-n-1} $.
3. For $ f(x) = 1/g(x) $, if $ g(a) \neq 0 $, then: $$\begin{align} \left( \frac{1}{g} \right)'(a) = -\frac{g'(a)}{g(a)^2}. \end{align}$$

Criterion for Extrema

[Extrema Criterion (29.1)] Suppose $ f $ is defined on an open interval $ I $ and has a maximum or minimum at $ x_0 \in{} I $. If $ f $ is differentiable at $ x_0 $, then: $$\begin{align} f'(x_0) = 0. \end{align}$$ Suppose $ f : [a, b] \to{} \mathbb{R} $ is continuous and attains its maximum or minimum at $ x_0 $. Then one of the following holds:

1. $ x_0 \in \{a, b\} $,
2. $ f $ is not differentiable at $ x_0 $,
3. $ f'(x_0) = 0 $.

Rolle’s Theorem

[Rolle’s Theorem (29.2)] Suppose $ f : [a, b] \to{} \mathbb{R} $ is continuous, differentiable on $ (a, b) $, and $ f(a) = f(b) $. Then there exists $ c \in{} (a, b) $ such that: $$\begin{align} f'(c) = 0. \end{align}$$

Mean Value Theorem

[Mean Value Theorem (29.3)] Suppose $ f : [a, b] \to{} \mathbb{R} $ is continuous, differentiable on $ (a, b) $. Then there exists $ c \in{} (a, b) $ such that: $$\begin{align} f'(c) = \frac{f(b) - f(a)}{b - a}. \end{align}$$

Examples and Consequences of MVT

[Constant Function (29.4)] If $ f $ is differentiable on $ (a, b) $ and $ f'(x) = 0 $ for all $ x \in{} (a, b) $, then $ f $ is a constant function. [Equality of Derivatives (29.5)] If $ f $ and $ g $ are differentiable on $ (a, b) $ and $ f'(x) = g'(x) $ for all $ x \in{} (a, b) $, then $ f(x) - g(x) = c $ for some constant $ c $.

Rolle’s Theorem, Mean Value Theorem, and Applications (Section 29)

Rolle’s Theorem

[Rolle’s Theorem (29.2)] Suppose $ f : [a, b] \to{} \mathbb{R} $ is continuous, differentiable on $ (a, b) $, and $ f(a) = f(b) $. Then there exists $ x \in{} (a, b) $ such that: $$\begin{align} f'(x) = 0. \end{align}$$

Mean Value Theorem

[Mean Value Theorem (29.3)] Suppose $ f : [a, b] \to{} \mathbb{R} $ is continuous, differentiable on $ (a, b) $. Then there exists $ x \in{} (a, b) $ such that: $$\begin{align} f'(x) = \frac{f(b) - f(a)}{b - a}. \end{align}$$
[MVT Application] For $ x, y \in{} \mathbb{R} $, $ |\sin{} x - \sin{} y| \leq{} |x - y| $. Apply MVT to $ f(t) = \sin{} t $ on $ [x, y] $: $ \exists{} z \in{} (x, y) $ such that: $$\begin{align} \frac{f(x) - f(y)}{x - y} = f'(z) = \cos z. \end{align}$$ Since $ |\cos{} z| \leq{} 1 $, $ | \frac{f(x) - f(y)}{x - y} | = |\cos{} z| \leq{} 1 $, hence $ |\sin{} x - \sin{} y| \leq{} |x - y| $.

Corollaries of MVT

[Constant Functions (29.4)] If $ f $ is differentiable on $ (a, b) $ and $ f' = 0 $ on $ (a, b) $, then $ f $ is constant.
[Equality of Derivatives (29.5)] If $ f, g $ are differentiable on $ (a, b) $ and $ f' = g' $ on $ (a, b) $, then $ \exists{} c \in{} \mathbb{R} $ such that $ f(x) - g(x) = c $ for all $ x \in{} (a, b) $.

Increasing and Decreasing Functions

[Monotonicity (29.6)] A function $ f $ on an interval $ I $ is:

1. {Increasing} if $ f(x_1) \leq f(x_2) $ for $ x_1 < x_2 $,
2. {Strictly increasing} if $ f(x_1) < f(x_2) $ for $ x_1 < x_2 $.

[Monotonicity and Derivatives (29.7)] Suppose $ f $ is differentiable on $ (a, b) $:

1. $ f $ is increasing if and only if $ f' \geq 0 $ on $ (a, b) $,
2. If $ f' > 0 $ on $ (a, b) $, then $ f $ is strictly increasing.

[Bernoulli’s Inequality] If $ n \in{} \mathbb{N} $ and $ x > -1 $, then: $$\begin{align} (1 + x)^n \geq 1 + nx. \end{align}$$ Let $ f(x) = (1 + x)^n - (1 + nx) $ and show $ f(x) \geq{} 0 $ for $ x > -1 $. By differentiating $ f(x) $, we conclude $ f(x) $ is increasing and achieves its minimum at $ x = 0 $, where $ f(0) = 0 $.

Differentiating Inverse Functions and Integration (Sections 29, 32)

Differentiating Inverse Functions

[Derivative of an Inverse Function (29.9)] Suppose $ I $ is an interval, $ f : I \to{} \mathbb{R} $ is a continuous, strictly monotone function. Let $ J = f(I) $, and $ g = f^{-1} : J \to{} I $. If $ f $ is differentiable at $ c \in{} I $, and $ f'(c) \neq{} 0 $, then $ g $ is differentiable at $ d = f(c) $, and: $$\begin{align} g'(d) = \frac{1}{f'(c)} = \frac{1}{f'(g(d))}. \end{align}$$

Derivatives of Rational Powers

[Derivative of Rational Powers] Let $ f(x) = x^n $ (strictly increasing on $ (0, \infty{}) $) with $ f'(x) = nx^{n-1} $. The inverse function is $ g(y) = y^{1/n} $. For $ y > 0 $: $$\begin{align} g'(y) = \frac{1}{f'(g(y))} = \frac{1}{n(y^{1/n})^{n-1}} = \frac{1}{n} y^{1/n - 1}. \end{align}$$ If $ n \in{} \mathbb{Z} $ is odd, extend $ f, g $ to $ \mathbb{R} $. Then $ g'(y) = \frac{1}{n} y^{1/n - 1} $ for $ y < 0 $. [Derivative of $ h(x) = x^r $, $ r \in{} \mathbb{Q} $] Write $ r = m/n $, $ h(x) = x^{m/n} $. Use the chain rule: $$\begin{align} h'(x) = \frac{m}{n} x^{r-1}. \end{align}$$

Inverse Trigonometric Functions

[Arcsine] For $ f(x) = \sin{} x $ on $ [-\pi{}/2, \pi{}/2] $, $ g = \arcsin{} : [-1, 1] \to{} [-\pi{}/2, \pi{}/2] $. Since $ f'(x) = \cos{} x $, for $ y \in{} (-1, 1) $: $$\begin{align} (\arcsin y)' = \frac{1}{\sqrt{1-y^2}}. \end{align}$$ [Arctangent] For $ f(x) = \tan{} x $ on $ (-\pi{}/2, \pi{}/2) $, $ g = \arctan{} : \mathbb{R} \to{} (-\pi{}/2, \pi{}/2) $. Since $ f'(x) = 1 + x^2 $, for $ y \in{} \mathbb{R} $: $$\begin{align} (\arctan y)' = \frac{1}{1 + y^2}. \end{align}$$

Integration: Concepts and Definitions

[Darboux Sums] Let $ f : [a, b] \to{} \mathbb{R} $ be bounded.

1. Partition $ P = \{t_0, t_1, \ldots, t_n\} $ of $ [a, b] $ gives subintervals $ [t_{k-1}, t_k] $.
2. Lower Darboux sum: $$\begin{align} L(f, P) = \sum_{k=1}^n m(f, [t_{k-1}, t_k])(t_k - t_{k-1}), \end{align}$$ where $ m(f, [t_{k-1}, t_k]) = \inf_{x \in [t_{k-1}, t_k]} f(x) $.
3. Upper Darboux sum: $$\begin{align} U(f, P) = \sum_{k=1}^n M(f, [t_{k-1}, t_k])(t_k - t_{k-1}), \end{align}$$ where $ M(f, [t_{k-1}, t_k]) = \sup_{x \in [t_{k-1}, t_k]} f(x) $.

[Integrability] $ f $ is integrable on $ [a, b] $ if: $$\begin{align} \sup_P L(f, P) = \inf_P U(f, P), \end{align}$$ denoted $ \int_a{}^b f(x) dx $.

Examples of Integrability

[Constant Function] If $ f(x) = c $, then: $$\begin{align} \int_a^b f(x) dx = c(b-a). \end{align}$$ [Discontinuous Function] Let $ g(x) = 1 $ if $ x \in{} \mathbb{Q} $, $ g(x) = 0 $ otherwise. Then: $$\begin{align} \sup_P L(g, P) = 0, \quad \inf_P U(g, P) = 1. \end{align}$$ Since $ \sup_P{} L \neq{} \inf_P{} U $, $ g $ is not integrable. [Linear Function] If $ h(x) = x $, then: $$\begin{align} \int_0^b h(x) dx = \frac{b^2}{2}. \end{align}$$

Darboux Sums, Integrability, and Riemann Integration (Sections 32-33)

Darboux Sums and Integrals

[Darboux Sums] Let $ f : [a, b] \to{} \mathbb{R} $ be bounded. For a partition $ P = \{a = t_0 < t_1 < \cdots{} < t_n = b\} $:

1. Lower Darboux sum: $$\begin{align} L(f, P) = \sum_{k=1}^n m(f, [t_{k-1}, t_k])(t_k - t_{k-1}), \end{align}$$ where $ m(f, [t_{k-1}, t_k]) = \inf_{x \in [t_{k-1}, t_k]} f(x) $.
2. Upper Darboux sum: $$\begin{align} U(f, P) = \sum_{k=1}^n M(f, [t_{k-1}, t_k])(t_k - t_{k-1}), \end{align}$$ where $ M(f, [t_{k-1}, t_k]) = \sup_{x \in [t_{k-1}, t_k]} f(x) $.

[Integrability] The lower Darboux integral is $ L(f) = \sup_P{} L(f, P) $, and the upper Darboux integral is $ U(f) = \inf_P{} U(f, P) $. $ f $ is integrable if $ L(f) = U(f) $, denoted: $$\begin{align} \int_a^b f = L(f) = U(f). \end{align}$$ [32.4] If $ f : [a, b] \to{} \mathbb{R} $ is bounded, then $ L(f) \leq{} U(f) $.

Integrals: Example

[Linear Function] Is $ h(x) = x $ integrable on $ [0, b] $? Compute $ \int_0{}^b h(x) dx $. For $ P = \{0, \frac{b}{n}, \frac{2b}{n}, \ldots{}, b\} $: $$\begin{align} L(h, P_n) = \frac{b^2}{2}\left(1 - \frac{1}{n}\right), \quad U(h, P_n) = \frac{b^2}{2}. \end{align}$$ Thus: $$\begin{align} L(h) \geq \sup_n L(h, P_n) = \frac{b^2}{2}, \quad U(h) \leq \lim_n U(h, P_n) = \frac{b^2}{2}. \end{align}$$ Since $ L(h) = U(h) $, $ h(x) $ is integrable with: $$\begin{align} \int_0^b h(x) dx = \frac{b^2}{2}. \end{align}$$

Criterion for Integrability

[32.5] A bounded function $ f : [a, b] \to{} \mathbb{R} $ is integrable if and only if for all $ \varepsilon{} > 0 $, there exists a partition $ P $ such that: $$\begin{align} U(f, P) - L(f, P) < \varepsilon. \end{align}$$

Monotone and Continuous Functions

[33.1] Any monotone function on $ [a, b] $ is integrable. [33.2] Any continuous function on $ [a, b] $ is integrable.

Mesh of a Partition

[Mesh (32.6)] The mesh of a partition $ P = \{t_0, t_1, \ldots{}, t_n\} $ is: $$\begin{align} \text{mesh}(P) = \max_{1 \leq k \leq n} (t_k - t_{k-1}). \end{align}$$ [32.7] A bounded $ f : [a, b] \to{} \mathbb{R} $ is integrable if and only if for all $ \varepsilon{} > 0 $, there exists $ \delta{} > 0 $ such that: $$\begin{align} U(f, P) - L(f, P) < \varepsilon \quad \text{whenever mesh}(P) < \delta. \end{align}$$

Riemann Integration

[Riemann Integral (32.8)] Let $ f : [a, b] \to{} \mathbb{R} $ be bounded. For a partition $ P = \{t_0, t_1, \ldots{}, t_n\} $ and $ x_k \in{} [t_{k-1}, t_k] $, define the Riemann sum: $$\begin{align} S = \sum_{k=1}^n f(x_k)(t_k - t_{k-1}). \end{align}$$ $ f $ is Riemann integrable if there exists $ r \in{} \mathbb{R} $ such that for all $ \varepsilon{} > 0 $, there exists $ \delta{} > 0 $ such that: $$\begin{align} |S - r| < \varepsilon \quad \text{whenever mesh}(P) < \delta. \end{align}$$ [32.9] A bounded function $ f : [a, b] \to{} \mathbb{R} $ is Riemann integrable if and only if it is Darboux integrable. In this case: $$\begin{align} \int_a^b f = R\int_a^b f. \end{align}$$

Integrability and Riemann Integration (Sections 32-33)

Monotone and Continuous Functions

[Integrability Criterion (32.5)] A bounded function $ f : [a, b] \to{} \mathbb{R} $ is integrable if and only if: $$\begin{align} \forall \varepsilon > 0, \exists \text{ a partition } P \text{ such that } U(f, P) - L(f, P) < \varepsilon. \end{align}$$ [Monotone Functions Are Integrable (33.1)] Any monotone function on $ [a, b] $ is integrable.
[Continuous Functions Are Integrable (33.2)] Any continuous function on $ [a, b] $ is integrable.

Mesh of a Partition

[Mesh (32.6)] The mesh of a partition $ P = \{t_0, t_1, \ldots{}, t_n\} $ is: $$\begin{align} \text{mesh}(P) = \max_{1 \leq k \leq n} (t_k - t_{k-1}). \end{align}$$ [Integrability and Mesh (32.7)] A bounded function $ f : [a, b] \to{} \mathbb{R} $ is integrable if and only if: $$\begin{align} \forall \varepsilon > 0, \exists \delta > 0 \text{ such that } U(f, P) - L(f, P) < \varepsilon \text{ whenever mesh}(P) < \delta. \end{align}$$

Riemann Integration

[Riemann Integral (32.8)] Let $ f : [a, b] \to{} \mathbb{R} $ be bounded. For a partition $ P = \{t_0, t_1, \ldots{}, t_n\} $ and $ x_k \in{} [t_{k-1}, t_k] $, define the Riemann sum: $$\begin{align} S = \sum_{k=1}^n f(x_k)(t_k - t_{k-1}). \end{align}$$ $ f $ is Riemann integrable if: $$\begin{align} \exists r \in \mathbb{R} \text{ such that } \forall \varepsilon > 0, \exists \delta > 0 \text{ such that } |S - r| < \varepsilon \text{ whenever mesh}(P) < \delta. \end{align}$$ [Equivalence of Riemann and Darboux Integrability (32.9)] A bounded $ f : [a, b] \to{} \mathbb{R} $ is Riemann integrable if and only if it is Darboux integrable. In this case: $$\begin{align} R\int_a^b f = \int_a^b f. \end{align}$$

Properties of Integrable Functions

[Exercise 32.7] If $ f $ is integrable on $ [a, b] $, and $ f = g $ except at finitely many points, then $ g $ is integrable on $ [a, b] $ and: $$\begin{align} \int_a^b f = \int_a^b g. \end{align}$$ The statement fails if the set of exceptions is countably infinite. For example: $$\begin{align} f(x) = 0, \quad g(x) = \begin{cases} 1 & x \in \mathbb{Q}, \\ 0 & x \notin \mathbb{Q}. \end{cases} \end{align}$$ $ f $ is integrable, but $ g $ is not.

Properties of Integrals and Convergence Theorems (Section 33)

Properties of Integrals

[Linearity and Comparison of Integrals (33.3, 33.4(i))] Suppose $ f, g $ are integrable on $ [a, b] $, and $ c \in{} \mathbb{R} $. Then:

1. $ cf $ is integrable, and: $$\begin{align} \int_a^b cf = c \int_a^b f. \end{align}$$
2. $ f + g $ is integrable, and: $$\begin{align} \int_a^b (f + g) = \int_a^b f + \int_a^b g. \end{align}$$
3. If $ f \geq g $, then: $$\begin{align} \int_a^b f \geq \int_a^b g. \end{align}$$

[Triangle Inequality for Integrals (33.5)] If $ f $ is integrable on $ [a, b] $, then $ |f| $ is integrable, and: $$\begin{align} \left| \int_a^b f \right| \leq \int_a^b |f|. \end{align}$$

Integrability of Products and Piecewise Functions

If $ f $ is integrable on $ [a, b] $, then $ f^2 $ is integrable. If $ f, g $ are integrable on $ [a, b] $, then $ fg $ is integrable.
[Piecewise Monotone and Continuous Functions (33.8)] Suppose $ f : [a, b] \to{} \mathbb{R} $ is either:

1. Piecewise monotone and bounded, or
2. Piecewise continuous.

Then $ f $ is integrable.

Convergence and Interchange of Limits and Integrals

Suppose $ (f_n) $ is a sequence of integrable functions on $ [a, b] $ that converges uniformly to $ f $. Then $ f $ is integrable, and: $$\begin{align} \int_a^b f = \lim_{n \to \infty} \int_a^b f_n. \end{align}$$

Convergence Theorems

[Bounded Convergence (33.11)] Suppose $ (f_n) $ are integrable on $ [a, b] $, $ |f_n| \leq{} M $ for all $ n $, $ f_n \to{} f $ pointwise on $ [a, b] $, and $ f $ is integrable. Then: $$\begin{align} \lim_{n \to \infty} \int_a^b f_n = \int_a^b f. \end{align}$$ [Monotone Convergence (33.12)] Suppose $ (f_n) $ are integrable on $ [a, b] $, $ f_1 \leq{} f_2 \leq{} \cdots{} $, $ f_n \to{} f $ pointwise on $ [a, b] $, and $ f $ is integrable. Then: $$\begin{align} \lim_{n \to \infty} \int_a^b f_n = \int_a^b f. \end{align}$$ [Application of Monotone Convergence] Let $ f_n(x) = \frac{1}{1 + nx^3} $ on $ [0, 1] $. Then: $$\begin{align} \int_0^1 f_n(x) dx \to \int_0^1 f(x) dx = 0, \end{align}$$ where $$\begin{align} f(x) = \begin{cases} 1, & x = 0, \\ 0, & x \in (0, 1]. \end{cases} \end{align}$$

Fundamental Theorems of Calculus and Change of Variable (Section 34)

Fundamental Theorem of Calculus I

[Fundamental Theorem of Calculus I (34.1)] Suppose $ g : [a, b] \to{} \mathbb{R} $ is continuous, differentiable on $ (a, b) $, and $ g' $ is integrable on $ [a, b] $. Then: $$\begin{align} \int_a^b g'(x) \, dx = g(b) - g(a). \end{align}$$ Compute $ \int_a{}^b x^n \, dx $. Use $ g(x) = \frac{x^{n+1}}{n+1} $, so: $$\begin{align} \int_a^b x^n \, dx = \frac{b^{n+1} - a^{n+1}}{n+1}. \end{align}$$

Integration by Parts

[Integration by Parts (34.2)] Suppose $ u, v : [a, b] \to{} \mathbb{R} $ are continuous, differentiable on $ (a, b) $, and $ u', v' $ are integrable on $ [a, b] $. Then: $$\begin{align} \int_a^b u(x)v'(x) \, dx + \int_a^b u'(x)v(x) \, dx = u(b)v(b) - u(a)v(a). \end{align}$$ Compute $ \int_0{}^\pi{} x \cos{} x \, dx $. Let $ u(x) = x $, $ v'(x) = \cos{} x $: $$\begin{align} \int_0^\pi x \cos x \, dx = x \sin x \big|_0^\pi - \int_0^\pi \sin x \, dx = 0 - (-2) = -2. \end{align}$$

Fundamental Theorem of Calculus II

[Fundamental Theorem of Calculus II (34.3)] Suppose $ f : [a, b] \to{} \mathbb{R} $ is integrable. Define: $$\begin{align} F(x) = \int_a^x f(t) \, dt. \end{align}$$ If $ f $ is continuous at $ c $, then $ F $ is differentiable at $ c $, with $ F'(c) = f(c) $. Let $ G(x) = \int_{x^2}^2 \sin{}(t^2) \, dt $. Then $ G'(x) = -2x \sin{}(x^4) $ by the Chain Rule.

Change of Variable in Integrals

[Change of Variable (34.4)] Suppose $ u : J \to{} I $, $ u' $ is continuous, and $ f : I \to{} \mathbb{R} $ is continuous. Then for $ a, b \in{} J $: $$\begin{align} \int_a^b f(u(x)) u'(x) \, dx = \int_{u(a)}^{u(b)} f(t) \, dt. \end{align}$$ Compute $ \int_1{}^4 \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx $. Let $ u(x) = \sqrt{x} $, then $ u'(x) = \frac{1}{2\sqrt{x}} $: $$\begin{align} \int_1^4 \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx = \int_1^2 2 \sin t \, dt = 2(-\cos t) \big|_1^2 = 2(\cos 1 - \cos 2). \end{align}$$

Interchanging Integration, Differentiation, and Power Series (Section 26)

Interchanging Integration with Limits and Sums

[Exercise 33.9, Lecture 32] Suppose $ (f_n) $ is a sequence of integrable functions on $ [a, b] $ converging uniformly to $ f $. Then: $$\begin{align} \lim_{n \to \infty} \int_a^b f_n = \int_a^b f. \end{align}$$ If $ g_n $ are integrable on $ [a, b] $, and $ f = \sum_{n=0}^\infty{} g_n $ converges uniformly, then $ f $ is integrable, and: $$\begin{align} \int_a^b f = \sum_{n=0}^\infty \int_a^b g_n. \end{align}$$

Interchanging Differentiation with Limits and Sums

Let $ f_n(x) = \frac{1}{n} \sin{}(n^2 x) $. Then $ f_n \to{} 0 $ uniformly on $ \mathbb{R} $. However: $$\begin{align} f_n'(x) = n \cos(n^2 x). \end{align}$$ If $ x = \frac{p}{q} \pi{} $, then $ f_n'(x) $ does not converge, even pointwise.

Power Series and Radius of Convergence

[Radius of Convergence] For a power series $ \sum_{n=0}^\infty{} a_n x^n $, let: $$\begin{align} \beta = \limsup_{n \to \infty} |a_n|^{1/n}. \end{align}$$ The radius of convergence is $ R = \frac{1}{\beta{}} $. [Uniform Convergence (26.1)] The series $ \sum_{n=0}^\infty{} a_n x^n $ converges uniformly on $ [-R_1, R_1] $ for $ R_1 < R $. The series $ \sum_{n=0}^\infty{} a_n x^n $ converges to a continuous function on $ (-R, R) $.

Differentiation and Integration of Power Series

[Differentiation and Integration (26.3)] If $ \sum_{n=0}^\infty{} a_n x^n $ has radius of convergence $ R $, then:

1. $ \sum_{n=1}^\infty n a_n x^{n-1} $ has radius of convergence $ R $,
2. $ \sum_{n=0}^\infty \frac{a_n}{n+1} x^{n+1} $ has radius of convergence $ R $.

[Integration of Power Series (26.4)] Suppose $ f(t) = \sum_{n=0}^\infty{} a_n t^n $ has radius of convergence $ R $. Then for $ |x| < R $: $$\begin{align} \int_0^x f(t) \, dt = \sum_{n=0}^\infty \frac{a_n}{n+1} x^{n+1}. \end{align}$$ For $ f(t) = \frac{1}{1-t} = \sum_{n=0}^\infty{} t^n $ ($ R = 1 $): $$\begin{align} -\ln(1-x) = \int_0^x f(t) \, dt = \sum_{n=0}^\infty \frac{x^{n+1}}{n+1}, \quad |x| < 1. \end{align}$$ [Differentiation of Power Series (26.5)] Suppose $ f(t) = \sum_{n=0}^\infty{} a_n t^n $ has radius of convergence $ R $. Then for $ |t| < R $: $$\begin{align} f'(t) = \sum_{n=1}^\infty n a_n t^{n-1}. \end{align}$$

Abel’s Theorem

[Abel’s Theorem (26.6)] Suppose $ f(x) = \sum_{n=0}^\infty{} a_n x^n $ has radius of convergence $ R > 0 $. If the series converges at $ R $ (or $ -R $), then $ f $ is continuous at $ R $ (or $ -R $).

Abel’s Theorem, Convexity, and Inequalities (Section 26)

Abel Summation Theorem

[Abel’s Theorem (26.6)] Let $ f(x) = \sum_{n=0}^\infty{} a_n x^n $ have radius of convergence $ R > 0 $. If the series converges at $ R $ (or $ -R $), then $ f $ is continuous at $ R $ (or $ -R $. $$\begin{align} 1 - \frac{1}{2} + \frac{1}{3} - \cdots = \ln 2. \end{align}$$ Let $ g(t) = \frac{1}{1+t} = \sum_{n=0}^\infty{} (-1)^n t^n $ ($ R = 1 $), and $ f(x) = \sum_{k=1}^\infty{} \frac{(-1)^{k-1}}{k} x^k $. The series diverges at $ -1 $ but converges at $ 1 $: $$\begin{align} \ln(1+x) = \int_0^x g(t) \, dt = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} x^k. \end{align}$$ Thus: $$\begin{align} f(1) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} = \ln 2. \end{align}$$

Alternating Series

[Alternating Series Test] Suppose $ a_1 \geq{} a_2 \geq{} \cdots{} \geq{} 0 $. Then: $$\begin{align} \sum_{k=1}^\infty (-1)^{k-1} a_k = a_1 - a_2 + a_3 - \cdots \end{align}$$ converges if and only if $ \lim_{k \to \infty} a_k = 0 $. $$\begin{align} 1 - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}. \end{align}$$ Let $ f(x) = \arctan{} x $, so $ f'(x) = \frac{1}{1+x^2} $. For $ |x| < 1 $: $$\begin{align} f'(x) = \sum_{n=0}^\infty (-1)^n x^{2n}, \quad f(x) = \int_0^x f'(t) \, dt = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1}. \end{align}$$ At $ x = 1 $: $$\begin{align} \frac{\pi}{4} = \arctan 1 = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}. \end{align}$$

Convex Functions

[Convexity] A function $ f $ on $ I $ is convex if: $$\begin{align} f\left(\frac{x+y}{2}\right) \leq \frac{f(x) + f(y)}{2}, \quad \forall x, y \in I. \end{align}$$ If $ f $ is convex, then: $$\begin{align} f((1-t)x + ty) \leq (1-t)f(x) + tf(y), \quad \forall t \in (0, 1). \end{align}$$

Criteria for Convexity

If $ f $ is differentiable on $ I $, and $ f' $ is increasing, then $ f $ is convex. If $ f $ is twice differentiable on $ I $, and $ f'' \geq{} 0 $, then $ f $ is convex.

1. $ f(x) = e^x $ is convex on $ \mathbb{R} $ because $ f''(x) = e^x > 0 $.
2. $ g(x) = \ln x $ is concave on $ (0, \infty) $ because $ g''(x) = -\frac{1}{x^2} < 0 $.

Inequalities

[Jensen’s Inequality] If $ f $ is convex on $ I $, $ x_1, \ldots{}, x_n \in{} I $, $ t_1, \ldots{}, t_n \geq{} 0 $, and $ \sum_{i=1}^n t_i = 1 $, then: $$\begin{align} f\left(\sum_{i=1}^n t_i x_i\right) \leq \sum_{i=1}^n t_i f(x_i). \end{align}$$ [Arithmetic-Geometric Means Inequality] If $ x_1, \ldots{}, x_n > 0 $, $ t_1, \ldots{}, t_n > 0 $, and $ \sum_{i=1}^n t_i = 1 $, then: $$\begin{align} \sum_{i=1}^n t_i x_i \geq \prod_{i=1}^n x_i^{t_i}. \end{align}$$ [Special Case] If $ x_1, \ldots{}, x_n > 0 $, then: $$\begin{align} \frac{x_1 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}. \end{align}$$

Convexity, Inequalities, and Nowhere Differentiable Functions (Section 36)

Jensen’s Inequality for Convex Functions

[Jensen’s Inequality] Let $ f $ be a convex function on an interval $ I $, and let $ x_1, \ldots{}, x_n \in{} I $ with $ t_1, \ldots{}, t_n \geq{} 0 $ and $ \sum_{i=1}^n t_i = 1 $. Then: $$\begin{align} f\left(\sum_{i=1}^n t_i x_i\right) \leq \sum_{i=1}^n t_i f(x_i). \end{align}$$ If $ f $ is concave, the inequality is reversed.

Inequalities Between Means

[Power Mean Inequality] Suppose $ r > 1 $ and $ x_1, \ldots{}, x_n \geq{} 0 $. Then: $$\begin{align} \frac{x_1 + \cdots + x_n}{n} \leq \left(\frac{x_1^r + \cdots + x_n^r}{n}\right)^{1/r}. \end{align}$$ If $ r = 2 $, this gives the inequality between arithmetic and quadratic means: $$\begin{align} \frac{x_1 + \cdots + x_n}{n} \leq \sqrt{\frac{x_1^2 + \cdots + x_n^2}{n}}. \end{align}$$

Arithmetic and Harmonic Means

[Arithmetic-Harmonic Mean Inequality] If $ x_1, \ldots{}, x_n > 0 $, then: $$\begin{align} \frac{x_1 + \cdots + x_n}{n} \geq \frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}}. \end{align}$$

Nowhere Differentiable Functions

There exists a bounded, uniformly continuous function $ f : \mathbb{R} \to{} \mathbb{R} $ that is differentiable nowhere.

Infinite Primes and Divergence of Series

Let $ p_1 < p_2 < \cdots{} $ be the increasing sequence of prime numbers. Then: $$\begin{align} \sum_{n=1}^\infty \frac{1}{p_n} \quad \text{diverges}. \end{align}$$