Real Functions and Their Graphs

Analysis studies functions of a real variable, and it studies them with a short list of questions. Where does the function increase? What is the largest value it takes on a set? Does it have an inverse? What happens when it is combined with the integer part of a number, which jumps at every integer? This chapter fixes the objects and answers every such question that does not need a limit.

A real function is a rule with a domain and a graph, and functions can be added, multiplied and composed. A strictly monotone function has an inverse, and the inverse is monotone too. When the values of a function on a non-empty set are bounded, they have a supremum, and there are rules for comparing suprema. The exponential, the logarithm and real powers are built from three facts about the exponential that Integration, Series, and Function Spaces proves. The last section collects strange functions on which later definitions of continuity and the integral will be tested.

Part III of this volume begins here. Its three chapters, 8, 9 and 10, are the tools that Analysis on the Real Line assumes on its first page. They use the algebra of Part II (Chapters 4–7) only a little, mostly for polynomials.

What you need.

• Functions, composition and inverse functions, from Sets and Functions, Chapter 2 (Sets and Functions, Definition 2.4.2).
• The supremum of an ordered set, from Sets and Functions, Chapter 5 (Sets and Functions, Definition 5.3.5).
• The real numbers as a complete ordered field, with the absolute value, the Archimedean property and the integer part, from Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$, Theorem 6.1.2).
• Roots and rational powers, from Chapter 1 (Definition 1.5.2).
• Polynomial functions, from Chapters 6 and 7, in Sections 8.1, 8.3 and 8.6 (Definition 6.4.1).

Time. About 30 hours if you read the main text and do every A exercise and one B and one C exercise per section; about 45 hours for everything.

Roadmap

• Real Functions and Their Graphs. A real function is a rule with a domain and a graph (Definition 8.1.1), and functions are added, multiplied, divided and composed.
• Monotone Functions and Inverse Functions. A strictly monotone function has an inverse that is monotone in the same direction (Theorem 8.2.3), which makes roots into functions.
• Bounded Functions and Their Suprema. The supremum of a function on a set, found by bounding every value and then finding a value close to the bound, and rules for comparing suprema (Theorem 8.3.3).
• Floor, Distance to the Integers, and Step Functions. The floor, the ceiling and the fractional part (Definition 8.4.1), the distance to the nearest integer, periodic functions and step functions.
• The Exponential, the Logarithm, and Real Powers. From three borrowed facts about the exponential come the logarithm and the powers $a^x$ for every real $x$, with the laws of powers (Proposition 8.5.6).
• A Table of Strange Functions. The Dirichlet function, Thomae’s function and other strange examples (Definition 8.6.2), each with a question that a later volume answers.