Chapter Review

Summary

A real function is a set of pairs with one value over each input (Definition 8.1.1), and the operations on functions are operations on values. A strictly monotone function is injective, reflects the order, and has a strictly monotone inverse (Theorem 8.2.3). The values of a function on a non-empty set have a supremum and an infimum when they are bounded. They are found by bounding every value and then producing a witness, and the rules of Theorem 8.3.3 compare them. The floor makes the integer part a function (Definition 8.4.1), and step functions are built from it. Two of the three borrowed facts about the exponential determine it (Theorem 8.5.3), and the logarithm is its inverse. The power $a^x$ is then defined for every positive $a$ and real $x$ . It agrees with the roots of Chapter 1 (Proposition 8.5.5) and obeys the laws of powers (Proposition 8.5.6).

What is open: continuity and limits of functions, which Analysis on the Real Line defines, and the integral, which Integration, Series, and Function Spaces defines; both test them on the functions of Section 8.6. Also open are the three borrowed facts about the exponential.

Proof methods in this chapter

Read a graph as a set of pairs (Example (Method: reading a graph as a set of pairs)). To move a graph by a bijection of the plane, fix a point $(x, y)$ and unwind “ $(x, y) \in \operatorname{graph} g$ ” into “ $x \in D_g$ and $y = g(x)$ ”. Introduced in Section 8.1.
Factor out $y - x$ (Example (Method: factor out y − x)). To show that $f$ is strictly increasing, write $f(y) - f(x) = (y - x)\,Q(x, y)$ and show $Q > 0$ for $x < y$ . Introduced in Section 8.2.
Bound every value, then find a witness (Example (Method: bound every value, then find a witness)). To prove $\sup_A f = M$ , show $f(x) \le M$ for every $x \in A$ . Then for each $\varepsilon > 0$ produce $x \in A$ with $f(x) > M - \varepsilon$ . Introduced in Section 8.3.
Pin the floor by its inequality (Example (Method: pin the floor by its inequality)). To show $\left\lfloor A \right\rfloor = m$ , check that $m$ is an integer with $m \le A < m + 1$ , or that $j \le A \iff j \le m$ for every integer $j$ . Introduced in Section 8.4.
Squeeze, then close the gap by Archimedes (Example (Method: squeeze, then close the gap by Archimedes)). To prove $\sup S = L$ , bound the set by $L$ and find elements within $C/n$ of it, with $C$ fixed before $n$ is chosen. Introduced in Section 8.5, where it proves the two-sided bound for the exponential.

Terms

bounded function: a function whose values on a set $A$ satisfy $\left| f(x) \right| \le M$ for one $M$ (Definition 8.3.1).
ceiling, floor, fractional part: $\left\lceil x \right\rceil$ , $\left\lfloor x \right\rfloor$ and $x - \left\lfloor x \right\rfloor$ (Definition 8.4.1).
Dirichlet function, Thomae’s function: the indicator of the rationals, and the function that is $1/q$ at $p/q$ in lowest terms and $0$ at irrational numbers (Definition 8.6.1, Definition 8.6.2).
even function, odd function: $f(-x) = f(x)$ and $f(-x) = -f(x)$ (Definition 8.1.4).
graph: the set $\{(x, f(x)) : x \in D\}$ (Definition 8.1.1).
logarithm: the inverse of the exponential, $\log \colon (0, \infty) \to \mathbb{R}$ (Definition 8.5.2).
monotone function: an increasing or decreasing function, strictly or not (Definition 8.2.1).
periodic function: a function that has a period $p \ne 0$ , with $f(x + p) = f(x)$ for every $x$ of its domain (Definition 8.4.3).
real power: $a^x$ for $a > 0$ and $x \in \mathbb{R}$ , defined through the exponential (Definition 8.5.3).
step function: a function on $[a, b]$ that is constant on each open piece of a partition $a = x_0 < \dots < x_n = b$ (Definition 8.4.4).
supremum of a function: the supremum of its values on a set, $\sup_A f$ (Definition 8.3.2).

Results used from earlier volumes

Logic and Proof:

from Chapter 4, induction (Logic and Proof, Axiom 4.2.1), in Sections 8.1, 8.5 and 8.6;
from Chapter 6, the negation of quantifiers (Logic and Proof, Proposition 6.2.1), in Section 8.2.

Sets and Functions:

from Chapter 1, ordered pairs, Cartesian products and relations (Sets and Functions, Definition 1.4.1, Sets and Functions, Definition 1.4.2, Sets and Functions, Definition 1.4.3);
from Chapter 2, functions (Sets and Functions, Definition 2.1.1), equality of functions (Sets and Functions, Proposition 2.1.1), the empty function (Sets and Functions, Proposition 2.1.2), images (Sets and Functions, Definition 2.5.1), composition and its associativity (Sets and Functions, Definition 2.1.2, Sets and Functions, Proposition 2.1.3), restrictions (Sets and Functions, Definition 2.3.1) and indicator functions (Sets and Functions, Definition 2.3.2);
from Chapter 2, inverses (Sets and Functions, Definition 2.4.2, Sets and Functions, Theorem 2.4.4), composites of injections and surjections (Sets and Functions, Theorem 2.2.1), and strictly monotone maps are injective (Sets and Functions, Proposition 2.2.3);
from Chapter 2, the examples that this chapter revisits (Sets and Functions, Example (Piecewise definitions are gluings), Sets and Functions, Example (Reflections, scalings and sums as images), Sets and Functions, Example (Maps from calculus), Sets and Functions, Example (Solving backwards, proving forwards), Sets and Functions, Example (Inverses found and checked));
from Chapter 3, induction in practice (Sets and Functions, Theorem 3.2.1), powers and their laws (Sets and Functions, Definition 3.3.2, Sets and Functions, Proposition 3.3.4), finite sums (Sets and Functions, Definition 3.3.1), Bernoulli’s inequality (Sets and Functions, Theorem 3.3.7) and the largest term of a finite list (Sets and Functions, Proposition 3.3.8);
from Chapter 4, finite sets and counting (Sets and Functions, Definition 4.1.2, Sets and Functions, Proposition 4.1.6, Sets and Functions, Proposition 4.1.5, Sets and Functions, Corollary 4.1.4), in Sections 8.4 and 8.6;
from Chapter 5, the supremum of an ordered set (Sets and Functions, Definition 5.3.5).

Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ :

from Chapter 2, commutative and ordered rings (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 2.3.2, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 2.4.1, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 2.4.2) and the rules of an ordered ring (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 2.4.3), on nearly every page;
from Chapter 3, reciprocals and order (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 3.2.2), the integer part of a rational (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 3.2.5) and the irrationality of $\sqrt 2$ (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 3.3.1);
from Chapter 4, the rules of an ordered field (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Lemma 4.1.1, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.1.2), the absolute value (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 4.1.1, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.1.3), the Archimedean property (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.2.3), the integer part (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.2.4), $\mathbb{Q}$ inside an ordered field and dense in it (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 4.2.2, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 4.2.5) and limits of sequences (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 4.3.2);
from Chapter 5, $\mathbb{R}$ as a complete ordered field (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 5.5.2, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 5.5.1) and two of its exercises that this chapter proves: infima by reflection (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Exercise (B1)), in Section 8.3, and the density of the irrationals (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Exercise (C2)), in Section 8.6;
from Chapter 6, square roots and the Archimedean property in a complete ordered field (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 6.1.4, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 6.1.2), and its uniqueness (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 6.3.1), for the borrowed exponential of Section 8.5.

Results used from earlier chapters

Chapter 1: power laws and the difference of powers in a commutative ring (Proposition 1.1.1, Lemma 1.1.8) and the rules for roots (Lemma 1.5.2), in Section 8.2; roots (Lemma 1.5.1), in Sections 8.2, 8.4, 8.5 and 8.6; the binomial theorem (Theorem 1.2.3), integer powers (Definition 1.5.1, Proposition 1.5.3) and rational powers (Definition 1.5.2), in Section 8.5.
Chapter 2: lowest terms of a fraction (Proposition 2.2.6), in Section 8.6.
Chapter 3: the real numbers inside the complex numbers (Proposition 3.1.1) and the modulus of a real number (Proposition 3.2.1), in Section 8.3; the borrowed cosine and sine (Axiom 3.5.2), for a comparison in Section 8.5.
Chapter 5: groups (Example (Groups)) and the subgroup test (Theorem 5.2.1), in Section 8.6.
Chapters 6 and 7, only for polynomials: evaluation as a function (Definition 6.4.1, Proposition 6.4.4), the identity theorem (Theorem 7.4.2) and the root count (Theorem 7.4.1), in Sections 8.1 and 8.6.
Chapter 7, for Section 8.3: the growth bound (Lemma 7.6.1) and the real root of an odd-degree polynomial (Corollary 7.6.11).

Chapter 4 is not needed.

Borrowed facts

Section 8.5 assumes three facts about the exponential function, Axiom 8.5.1, and proves everything else about the exponential, the logarithm and real powers from them. Nothing in Sections 8.1–8.4 uses these facts, and neither does the part of Section 8.5 before them. Integration, Series, and Function Spaces, Chapter 7 constructs the exponential and proves the three facts.

Threads

Order and inverses. The inverse of a strictly monotone function gives roots as functions in Section 8.2. Analysis on the Real Line proves that a continuous injective function on an interval is strictly monotone. Section 8.2 shows why both hypotheses are needed.
Suprema of functions. The rules of Section 8.3 are used again for the spread of values. Integration, Series, and Function Spaces calls it the oscillation and measures integrability with it.
Functions that resist drawing. The functions of Section 8.6 and the distance to the integers of Section 8.4 are where the analysis volumes test continuity, differentiability and the integral. A function that is continuous everywhere and differentiable nowhere is built there from tents of the form $\operatorname{dist}(2^k x, \mathbb{Z})/2^k$ . The zigzag of Section 8.6 is replaced by the sine once the sine exists.
The exponential. The exponential, borrowed in Section 8.5, is constructed in Integration, Series, and Function Spaces, Chapter 7. The sine and cosine borrowed in Section 3.5 are constructed there as well.