Section recaps

Each card collects the definitions and results of one section, plus the proof techniques that show up in its proofs. Click a heading to jump to the statement in the text. For a dense, printable version see the cheat sheet.

§2 The Set ℚ of Rational Numbers

Definitions

2.1 Definition

A number is called an algebraic number if it satisfies a polynomial equation

$$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 = 0$$

where the coefficients $c_0, c_1, \dots, c_n$ are integers, $c_n \ne 0$ and $n \ge 1$.

Key results

2.2 Rational Zeros Theorem

Suppose $c_0, c_1, \dots, c_n$ are integers and $r$ is a rational number satisfying

$$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 = 0 \tag{1}$$

where $n \ge 1$, $c_n \ne 0$ and $c_0 \ne 0$. Write $r = \frac cd$ where $c, d$ are integers with no common factors and $d \ne 0$. Then $c$ divides $c_0$ and $d$ divides $c_n$.

2.3 Corollary

Consider the polynomial equation $x^n + c_{n-1}x^{n-1} + \cdots + c_1 x + c_0 = 0$, where $c_0, \dots, c_{n-1}$ are integers and $c_0 \ne 0$. Any rational solution of this equation must be an integer that divides $c_0$.

§3 The Set ℝ of Real Numbers

Definitions

3.3 Definition

$|a| = a$ if $a \ge 0$, and $|a| = -a$ if $a \le 0$. $|a|$ is the absolute value of $a$.

3.4 Definition

For numbers $a$ and $b$, $\operatorname{dist}(a, b) = |a - b|$ is the distance between $a$ and $b$.

Key results

3.1 Theorem

The following are consequences of the field properties, for $a, b, c \in \mathbb{R}$:

1. $a + c = b + c$ implies $a = b$;
2. $a \cdot 0 = 0$ for all $a$;
3. $(-a)b = -ab$ for all $a, b$;
4. $(-a)(-b) = ab$ for all $a, b$;
5. $ac = bc$ and $c \ne 0$ imply $a = b$;
6. $ab = 0$ implies either $a = 0$ or $b = 0$.

3.2 Theorem

The following are consequences of the properties of an ordered field, for $a, b, c \in \mathbb{R}$:

1. If $a \le b$, then $-b \le -a$;
2. If $a \le b$ and $c \le 0$, then $bc \le ac$;
3. If $0 \le a$ and $0 \le b$, then $0 \le ab$;
4. $0 \le a^2$ for all $a$;
5. $0 < 1$;
6. If $0 < a$, then $0 < a^{-1}$;
7. If $0 < a < b$, then $0 < b^{-1} < a^{-1}$.

Here $a < b$ means $a \le b$ and $a \ne b$.

3.5 Theorem

1. $|a| \ge 0$ for all $a \in \mathbb{R}$.
2. $|ab| = |a| \cdot |b|$ for all $a, b \in \mathbb{R}$.
3. $|a + b| \le |a| + |b|$ for all $a, b \in \mathbb{R}$.

3.6 Corollary

$\operatorname{dist}(a, c) \le \operatorname{dist}(a, b) + \operatorname{dist}(b, c)$ for all $a, b, c \in \mathbb{R}$.

3.7 Triangle Inequality

$|a + b| \le |a| + |b|$ for all $a, b$.

Typical proof moves

• Triangle inequality to split an error
• Proof by contradiction
• Split into cases

§4 The Completeness Axiom

Definitions

4.1 Definition

Let $S$ be a nonempty subset of $\mathbb{R}$.

1. If $S$ contains a largest element $s_0$ (that is, $s_0 \in S$ and $s \le s_0$ for all $s \in S$), then $s_0$ is the maximum of $S$, written $s_0 = \max S$.
2. If $S$ contains a smallest element, it is the minimum of $S$, written $\min S$.

4.2 Definition

Let $S$ be a nonempty subset of $\mathbb{R}$.

1. If $M \in \mathbb{R}$ satisfies $s \le M$ for all $s \in S$, then $M$ is an upper bound of $S$, and $S$ is bounded above.
2. If $m \in \mathbb{R}$ satisfies $m \le s$ for all $s \in S$, then $m$ is a lower bound of $S$, and $S$ is bounded below.
3. $S$ is bounded if it is bounded above and below, i.e. $S \subseteq [m, M]$ for some real $m, M$.

4.3 Definition

Let $S$ be a nonempty subset of $\mathbb{R}$.

1. If $S$ is bounded above and has a least upper bound, we call it the supremum of $S$, written $\sup S$.
2. If $S$ is bounded below and has a greatest lower bound, we call it the infimum of $S$, written $\inf S$.

Key results

4.4 Completeness Axiom

Every nonempty subset $S$ of $\mathbb{R}$ that is bounded above has a least upper bound. In other words, $\sup S$ exists and is a real number.

4.5 Corollary

Every nonempty subset $S$ of $\mathbb{R}$ that is bounded below has a greatest lower bound $\inf S$.

4.6 Archimedean Property

If $a > 0$ and $b > 0$, then $na > b$ for some positive integer $n$.

4.7 Denseness of ℚ

If $a, b \in \mathbb{R}$ and $a < b$, then there is a rational $r \in \mathbb{Q}$ with $a < r < b$.

Typical proof moves

• Bound with a sup / inf (least upper bound property)
• Archimedean property to get N > 1/ε

§7 Limits of Sequences

Definitions

7.1 Definition

A sequence $(s_n)$ of real numbers converges to the real number $s$ if

$$\begin{gathered} \text{for each } \varepsilon > 0 \text{ there exists a number } N \text{ such that} \\ n > N \quad\text{implies}\quad |s_n - s| < \varepsilon. \end{gathered} \tag{1}$$

We then write $\lim_{n\to\infty} s_n = s$ or $s_n \to s$, and call $s$ the limit of $(s_n)$. A sequence that does not converge to any real number diverges.

§9 Limit Theorems for Sequences

Definitions

9.8 Definition

For a sequence $(s_n)$, we write $\lim s_n = +\infty$ provided

$$\text{for each } M > 0 \text{ there is a number } N \text{ such that } n > N \text{ implies } s_n > M.$$

We then say $(s_n)$ diverges to $+\infty$. Similarly, $\lim s_n = -\infty$ provided for each $M < 0$ there is $N$ such that $n > N$ implies $s_n < M$.

Key results

9.1 Theorem

Convergent sequences are bounded.

9.2 Theorem

If $(s_n)$ converges to $s$ and $k \in \mathbb{R}$, then $(ks_n)$ converges to $ks$. That is, $\lim(ks_n) = k \cdot \lim s_n$.

9.3 Theorem

If $(s_n)$ converges to $s$ and $(t_n)$ converges to $t$, then $(s_n + t_n)$ converges to $s + t$. That is, $\lim(s_n + t_n) = \lim s_n + \lim t_n$.

9.4 Theorem

If $(s_n)$ converges to $s$ and $(t_n)$ converges to $t$, then $(s_n t_n)$ converges to $st$. That is, $\lim(s_n t_n) = (\lim s_n)(\lim t_n)$.

9.5 Lemma

If $(s_n)$ converges to $s$, $s_n \ne 0$ for all $n$, and $s \ne 0$, then $(1/s_n)$ converges to $1/s$.

9.6 Theorem

Suppose $(s_n)$ converges to $s$ and $(t_n)$ converges to $t$. If $s \ne 0$ and $s_n \ne 0$ for all $n$, then $(t_n/s_n)$ converges to $t/s$.

9.7 Theorem (Basic Examples)

1. $\lim_{n\to\infty} \dfrac1{n^p} = 0$ for $p > 0$.
2. $\lim_{n\to\infty} a^n = 0$ if $|a| < 1$.
3. $\lim n^{1/n} = 1$.
4. $\lim_{n\to\infty} a^{1/n} = 1$ for $a > 0$.

9.9 Theorem

Let $(s_n)$ and $(t_n)$ be sequences with $\lim s_n = +\infty$ and $\lim t_n > 0$ ($\lim t_n$ may be finite or $+\infty$). Then $\lim s_n t_n = +\infty$.

9.10 Theorem

For a sequence $(s_n)$ of positive real numbers, $\lim s_n = +\infty$ if and only if $\lim \left(\dfrac1{s_n}\right) = 0$.

Typical proof moves

• Let ε > 0 be given, then find N (or δ)
• Pick N = max of several thresholds
• Triangle inequality to split an error
• Use ε/2 (or ε/3) so the pieces add up to ε
• Split into cases

§10 Monotone Sequences and Cauchy Sequences

Definitions

10.1 Definition

A sequence $(s_n)$ is increasing if $s_n \le s_{n+1}$ for all $n$, and decreasing if $s_n \ge s_{n+1}$ for all $n$. If $(s_n)$ is increasing then $s_n \le s_m$ whenever $n < m$. A sequence that is increasing or decreasing is monotone (or monotonic).

10.6 Definition

Let $(s_n)$ be a sequence in $\mathbb{R}$. Define

$$\limsup s_n = \lim_{N\to\infty} \sup\{s_n : n > N\} \tag{1}$$

$$\liminf s_n = \lim_{N\to\infty} \inf\{s_n : n > N\}. \tag{2}$$

If $(s_n)$ is not bounded above, $\sup\{s_n : n > N\} = +\infty$ for all $N$ and we decree $\limsup s_n = +\infty$. Likewise $\liminf s_n = -\infty$ if $(s_n)$ is not bounded below.

10.8 Definition

A sequence $(s_n)$ of real numbers is a Cauchy sequence if

$$\text{for each } \varepsilon > 0 \text{ there exists a number } N \text{ such that } m, n > N \text{ implies } |s_n - s_m| < \varepsilon. \tag{1}$$

Key results

10.2 Theorem

All bounded monotone sequences converge.

10.4 Theorem

1. If $(s_n)$ is an unbounded increasing sequence, then $\lim s_n = +\infty$.
2. If $(s_n)$ is an unbounded decreasing sequence, then $\lim s_n = -\infty$.

10.5 Corollary

If $(s_n)$ is monotone, then it either converges, diverges to $+\infty$, or diverges to $-\infty$. Thus $\lim s_n$ is always meaningful for monotone sequences.

10.7 Theorem

Let $(s_n)$ be a sequence in $\mathbb{R}$.

1. If $\lim s_n$ is defined (as a real number, $+\infty$ or $-\infty$), then $\liminf s_n = \lim s_n = \limsup s_n$.
2. If $\liminf s_n = \limsup s_n$, then $\lim s_n$ is defined and $\lim s_n = \liminf s_n = \limsup s_n$.

10.9 Lemma

Convergent sequences are Cauchy sequences.

10.10 Lemma

Cauchy sequences are bounded.

10.11 Theorem

A sequence is a convergent sequence if and only if it is a Cauchy sequence.

Typical proof moves

• Let ε > 0 be given, then find N (or δ)
• Triangle inequality to split an error
• Use ε/2 (or ε/3) so the pieces add up to ε
• Bound with a sup / inf (least upper bound property)
• Cauchy criterion (no limit needed)
• Split into cases

§11 Subsequences

Definitions

11.1 Definition

Suppose $(s_n)_{n\in\mathbb{N}}$ is a sequence. A subsequence of it is a sequence $(t_k)_{k\in\mathbb{N}}$ where for each $k$ there is a positive integer $n_k$ such that

$$n_1 < n_2 < \cdots < n_k < n_{k+1} < \cdots \tag{1}$$

$$t_k = s_{n_k}. \tag{2}$$

11.6 Definition

Let $(s_n)$ be a sequence in $\mathbb{R}$. A subsequential limit is any real number, or symbol $+\infty$ or $-\infty$, that is the limit of some subsequence of $(s_n)$.

Key results

11.2 Theorem

Let $(s_n)$ be a sequence.

1. If $t \in \mathbb{R}$, then there is a subsequence of $(s_n)$ converging to $t$ if and only if the set $\{n \in \mathbb{N} : |s_n - t| < \varepsilon\}$ is infinite for all $\varepsilon > 0$.
2. If $(s_n)$ is unbounded above, it has a subsequence with limit $+\infty$.
3. Similarly, if $(s_n)$ is unbounded below, a subsequence has limit $-\infty$.

In each case the subsequence can be taken to be monotonic.

11.3 Theorem

If $(s_n)$ converges, then every subsequence converges to the same limit.

11.4 Theorem

Every sequence $(s_n)$ has a monotonic subsequence.

11.5 Bolzano–Weierstrass Theorem

Every bounded sequence has a convergent subsequence.

11.7 Theorem

Let $(s_n)$ be any sequence. There is a monotonic subsequence whose limit is $\limsup s_n$, and a monotonic subsequence whose limit is $\liminf s_n$.

11.8 Theorem

Let $(s_n)$ be any sequence in $\mathbb{R}$ and let $S$ be its set of subsequential limits.

1. $S$ is nonempty.
2. $\sup S = \limsup s_n$ and $\inf S = \liminf s_n$.
3. $\lim s_n$ exists if and only if $S$ has exactly one element, namely $\lim s_n$.

11.9 Theorem

Let $S$ be the set of subsequential limits of $(s_n)$. If $(t_n)$ is a sequence in $S \cap \mathbb{R}$ and $t = \lim t_n$, then $t \in S$.

Typical proof moves

• Let ε > 0 be given, then find N (or δ)
• Proof by contradiction
• Induction
• Bound with a sup / inf (least upper bound property)
• Pass to a subsequence
• Use the sequential characterisation
• Split into cases

§12 lim sup's and lim inf's

Key results

12.1 Theorem

If $(s_n)$ converges to a positive real number $s$ and $(t_n)$ is any sequence, then

$$\limsup s_n t_n = s \cdot \limsup t_n.$$

Here we allow the conventions $s \cdot (+\infty) = +\infty$ and $s \cdot (-\infty) = -\infty$ for $s > 0$.

12.2 Theorem

Let $(s_n)$ be any sequence of nonzero real numbers. Then

$$\liminf \left|\frac{s_{n+1}}{s_n}\right| \le \liminf |s_n|^{1/n} \le \limsup |s_n|^{1/n} \le \limsup \left|\frac{s_{n+1}}{s_n}\right|.$$

12.3 Corollary

If $\lim\left|\frac{s_{n+1}}{s_n}\right|$ exists and equals $L$, then $\lim |s_n|^{1/n}$ exists and equals $L$.

Typical proof moves

• Bound with a sup / inf (least upper bound property)
• Pass to a subsequence
• Use the sequential characterisation
• Split into cases

§13 ★ Some Topological Concepts in Metric Spaces

Definitions

13.1 Definition

Let $S$ be a set and $d$ a function defined for all pairs $(x, y)$ of elements of $S$ satisfying

- D1. $d(x, x) = 0$ for all $x \in S$, and $d(x, y) > 0$ for distinct $x, y \in S$. - D2. $d(x, y) = d(y, x)$ for all $x, y \in S$. - D3. $d(x, z) \le d(x, y) + d(y, z)$ for all $x, y, z \in S$.

Such a $d$ is a distance function or metric on $S$. A metric space is a set $S$ together with a metric; properly speaking it is the pair $(S, d)$, since one set can carry several metrics (Exercise 13.1).

13.2 Definition

A sequence $(s_n)$ in a metric space $(S, d)$ converges to $s \in S$ if $\lim_{n\to\infty} d(s_n, s) = 0$. It is a Cauchy sequence if for each $\varepsilon > 0$ there is $N$ such that $m, n > N$ implies $d(s_m, s_n) < \varepsilon$. $(S, d)$ is complete if every Cauchy sequence in $S$ converges to some element of $S$.

13.6 Definition

Let $(S, d)$ be a metric space and $E \subseteq S$. A point $s_0 \in E$ is interior to $E$ if for some $r > 0$, $\{s \in S : d(s, s_0) < r\} \subseteq E$. $E^\circ$ denotes the set of interior points of $E$. $E$ is open in $S$ if every point of $E$ is interior to $E$, i.e. $E = E^\circ$.

13.8 Definition

A subset $E$ of a metric space $(S, d)$ is closed if its complement $S \setminus E$ is open. Because of 13.7(iii), any intersection of closed sets is closed (Exercise 13.5). The closure $E^-$ of $E$ is the intersection of all closed sets containing $E$. The boundary of $E$ is $E^- \setminus E^\circ$; its points are boundary points of $E$.

13.11 Definition

A family $\mathcal{U}$ of open sets is an open cover of $E$ if every point of $E$ lies in at least one set of $\mathcal{U}$, i.e. $E \subseteq \bigcup\{U : U \in \mathcal{U}\}$. A subcover is a subfamily that still covers $E$. A cover or subcover is finite if it has finitely many sets (the sets themselves may be infinite). $E$ is compact if every open cover of $E$ has a finite subcover of $E$.

Key results

13.3 Lemma

A sequence $(x^{(n)})$ in $\mathbb{R}^k$ converges if and only if each coordinate sequence $(x_j^{(n)})$ converges in $\mathbb{R}$, for $j = 1, \dots, k$. It is Cauchy in $\mathbb{R}^k$ if and only if each $(x_j^{(n)})$ is Cauchy in $\mathbb{R}$.

13.4 Theorem

Euclidean $k$-space $\mathbb{R}^k$ is complete.

13.5 Bolzano–Weierstrass Theorem

Every bounded sequence in $\mathbb{R}^k$ has a convergent subsequence.

13.9 Proposition

Let $E$ be a subset of a metric space $(S, d)$.

1. $E$ is closed if and only if $E = E^-$.
2. $E$ is closed if and only if it contains the limit of every convergent sequence of points in $E$.
3. A point is in $E^-$ if and only if it is the limit of some sequence of points in $E$.
4. A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.

13.10 Theorem

Let $(F_n)$ be a decreasing sequence $F_1 \supseteq F_2 \supseteq \cdots$ of closed, bounded, nonempty sets in $\mathbb{R}^k$. Then $F = \bigcap_{n=1}^\infty F_n$ is also closed, bounded and nonempty.

13.12 Heine–Borel Theorem

A subset $E$ of $\mathbb{R}^k$ is compact if and only if it is closed and bounded.

13.13 Proposition

Every $k$-cell $F$ in $\mathbb{R}^k$ is compact.

Typical proof moves

• Pick N = max of several thresholds
• Proof by contradiction
• Pass to a subsequence
• Cauchy criterion (no limit needed)

§14 Series

Definitions

14.1 Summation Notation

$\sum_{k=m}^n a_k$ is shorthand for $a_m + a_{m+1} + \cdots + a_n$. For example,

$$\sum_{k=2}^5 \frac1{k^2 + k} = \frac16 + \frac1{12} + \frac1{20} + \frac1{30}, \qquad \sum_{k=0}^n 2^{-k} = 1 + \frac12 + \frac14 + \cdots + \frac1{2^n}.$$

$\sum_{n=m}^\infty a_n$ is shorthand for $a_m + a_{m+1} + a_{m+2} + \cdots$, which we now give a meaning.

14.2 Infinite Series

The partial sums of $\sum_{n=m}^\infty a_n$ are $s_n = a_m + a_{m+1} + \cdots + a_n = \sum_{k=m}^n a_k$. The series converges if $(s_n)$ converges to a real number $S$, and then we define $\sum_{n=m}^\infty a_n = S$. Thus

$$\sum_{n=m}^\infty a_n = S \quad\text{means}\quad \lim s_n = S, \quad\text{i.e.}\quad \lim_{n\to\infty} \sum_{k=m}^n a_k = S.$$

A series that does not converge diverges. We write $\sum a_n = +\infty$ if $\lim s_n = +\infty$, and similarly for $-\infty$. The symbol $\sum_{n=m}^\infty a_n$ has no meaning otherwise. When the starting index does not matter we just write $\sum a_n$.

14.3 Definition

A series $\sum a_n$ satisfies the Cauchy criterion if its partial sums form a Cauchy sequence:

$$\text{for each } \varepsilon > 0 \text{ there is } N \text{ such that } m, n > N \text{ implies } |s_n - s_m| < \varepsilon. \tag{1}$$

We lose nothing by requiring $n > m$, and writing $m - 1$ in place of $m$, (1) is equivalent to

$$\text{for each } \varepsilon > 0 \text{ there is } N \text{ such that } n \ge m > N \text{ implies } |s_n - s_{m-1}| < \varepsilon. \tag{2}$$

Since $s_n - s_{m-1} = \sum_{k=m}^n a_k$, this is

$$\text{for each } \varepsilon > 0 \text{ there is } N \text{ such that } n \ge m > N \text{ implies } \left| \sum_{k=m}^n a_k \right| < \varepsilon. \tag{3}$$

We usually use version (3).

Key results

14.4 Theorem

A series converges if and only if it satisfies the Cauchy criterion.

14.5 Corollary

If a series $\sum a_n$ converges, then $\lim a_n = 0$.

14.6 Comparison Test

Let $\sum a_n$ be a series with $a_n \ge 0$ for all $n$.

1. If $\sum a_n$ converges and $|b_n| \le a_n$ for all $n$, then $\sum b_n$ converges.
2. If $\sum a_n = +\infty$ and $b_n \ge a_n$ for all $n$, then $\sum b_n = +\infty$.

14.7 Corollary

Absolutely convergent series are convergent.

14.8 Ratio Test

A series $\sum a_n$ of nonzero terms

1. converges absolutely if $\limsup |a_{n+1}/a_n| < 1$,
2. diverges if $\liminf |a_{n+1}/a_n| > 1$.
3. Otherwise $\liminf |a_{n+1}/a_n| \le 1 \le \limsup |a_{n+1}/a_n|$ and the test gives no information.

14.9 Root Test

Let $\sum a_n$ be a series and $\alpha = \limsup |a_n|^{1/n}$. The series $\sum a_n$

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

Typical proof moves

• Bound with a sup / inf (least upper bound property)
• Pass to a subsequence
• Compare with a known series or sequence
• Telescoping or geometric sum
• Cauchy criterion (no limit needed)
• Choose an explicit witness

§15 Alternating Series and Integral Tests

Key results

15.1 Theorem

$\sum \dfrac1{n^p}$ converges if and only if $p > 1$.

15.3 Alternating Series Theorem

If $a_1 \ge a_2 \ge \cdots \ge a_n \ge \cdots \ge 0$ and $\lim a_n = 0$, then the alternating series $\sum (-1)^{n+1} a_n$ converges. Moreover, the partial sums $s_n = \sum_{k=1}^n (-1)^{k+1} a_k$ satisfy $|s - s_n| \le a_n$ for all $n$.

Typical proof moves

• Compare with a known series or sequence

§16 ★ Decimal Expansions of Real Numbers

Definitions

16.4 Definition

An expression $K.d_1 d_2 \cdots d_\ell \overline{d_{\ell+1} \cdots d_{\ell+r}}$ represents the decimal expansion in which the block $d_{\ell+1} \cdots d_{\ell+r}$ repeats forever:

$$K.d_1 d_2 \cdots d_\ell\, d_{\ell+1} \cdots d_{\ell+r}\, d_{\ell+1} \cdots d_{\ell+r}\, d_{\ell+1} \cdots d_{\ell+r} \cdots.$$

Such an expansion is a repeating decimal.

Key results

16.2 Theorem

Every nonnegative real number $x$ has at least one decimal expansion.

16.3 Theorem

A real number $x$ has exactly one decimal expansion, or else $x$ has two decimal expansions, one ending in all $0$'s and the other ending in all $9$'s.

16.5 Theorem

A real number $x$ is rational if and only if its decimal expansion is repeating. (Theorem 16.3 shows that if $x$ has two decimal expansions, both are repeating.)

Typical proof moves

• Proof by contradiction
• Induction
• Split into cases

§17 Continuous Functions

Definitions

17.1 Definition

Let $f$ be a real-valued function whose domain is a subset of $\mathbb{R}$. The function $f$ is continuous at $x_0$ in $\operatorname{dom}(f)$ if, for every sequence $(x_n)$ in $\operatorname{dom}(f)$ converging to $x_0$, we have $\lim_n f(x_n) = f(x_0)$. If $f$ is continuous at each point of a set $S \subseteq \operatorname{dom}(f)$, then $f$ is continuous on $S$. The function $f$ is continuous if it is continuous on $\operatorname{dom}(f)$.

Key results

17.2 Theorem

Let $f$ be a real-valued function whose domain is a subset of $\mathbb{R}$. Then $f$ is continuous at $x_0$ in $\operatorname{dom}(f)$ if and only if

$$\begin{gathered} \text{for each } \varepsilon > 0 \text{ there exists } \delta > 0 \text{ such that} \\ x \in \operatorname{dom}(f) \text{ and } |x - x_0| < \delta \text{ imply } |f(x) - f(x_0)| < \varepsilon. \end{gathered} \tag{1}$$

17.3 Theorem

Let $f$ be a real-valued function with $\operatorname{dom}(f) \subseteq \mathbb{R}$. If $f$ is continuous at $x_0$ in $\operatorname{dom}(f)$, then $|f|$ and $kf$, $k \in \mathbb{R}$, are continuous at $x_0$.

17.4 Theorem

Let $f$ and $g$ be real-valued functions that are continuous at $x_0$ in $\mathbb{R}$. Then

1. $f + g$ is continuous at $x_0$;
2. $fg$ is continuous at $x_0$;
3. $f/g$ is continuous at $x_0$ if $g(x_0) \ne 0$.

17.5 Theorem

If $f$ is continuous at $x_0$ and $g$ is continuous at $f(x_0)$, then the composite function $g \circ f$ is continuous at $x_0$.

Typical proof moves

• Let ε > 0 be given, then find N (or δ)

§18 Properties of Continuous Functions

Key results

18.1 Theorem

Let $f$ be a continuous real-valued function on a closed interval $[a, b]$. Then $f$ is a bounded function. Moreover, $f$ assumes its maximum and minimum values on $[a, b]$: there exist $x_0, y_0$ in $[a, b]$ such that $f(x_0) \le f(x) \le f(y_0)$ for all $x \in [a, b]$.

18.2 Intermediate Value Theorem

If $f$ is a continuous real-valued function on an interval $I$, then $f$ has the intermediate value property on $I$: whenever $a, b \in I$, $a < b$, and $y$ lies between $f(a)$ and $f(b)$ (i.e. $f(a) < y < f(b)$ or $f(b) < y < f(a)$), there exists at least one $x$ in $(a, b)$ such that $f(x) = y$.

18.3 Corollary

If $f$ is a continuous real-valued function on an interval $I$, then the set $f(I) = \{f(x) : x \in I\}$ is also an interval or a single point.

18.4 Theorem

Let $f$ be a continuous strictly increasing function on an interval $I$. Then $f(I)$ is an interval $J$ by Corollary 18.3, and $f^{-1}$ is a function with domain $J$. The function $f^{-1}$ is continuous and strictly increasing on $J$.

18.5 Theorem

Let $g$ be a strictly increasing function on an interval $J$ such that $g(J)$ is an interval $I$. Then $g$ is continuous on $J$.

18.6 Theorem

Let $f$ be a one-to-one continuous function on an interval $I$. Then $f$ is strictly increasing ($x_1 < x_2$ implies $f(x_1) < f(x_2)$) or strictly decreasing ($x_1 < x_2$ implies $f(x_1) > f(x_2)$).

Typical proof moves

• Let ε > 0 be given, then find N (or δ)
• Proof by contradiction
• Bound with a sup / inf (least upper bound property)
• Pass to a subsequence
• Use the sequential characterisation
• Split into cases