Valued Functions

These notes provide an overview of functions in ${\reals.}$

Preliminaries
Intervals
Properties of Valued Functions
1. Monotonicity
2. Symmetry

Preliminaries

We begin with a brief review of functions. We've seen functions in the context of set theory, but we now focus specifically on functions in the context of real analysis.

valued function. A valued function is a mapping ${f: X \mapsto Y}$ where ${X \subseteq \reals}$ and ${Y \subseteq \reals,}$ such that, for every ${x \in D,}$ there exists a unique ${f(x) \in Y.}$ We call ${X}$ the domain of ${f,}$ that ${\reals}$ is the codomain of ${f,}$ and that ${Y}$ is the range of ${f.}$ By convention, we may denote a function in the form ${f(x) = \E,}$ where ${\E}$ is a predicate called a procedure. We say that ${f(x)}$ is the outcome of ${f,}$ that arguments are passed to a parameter of ${f}$ denoted ${x,}$ and that the function ${f}$ bears the name " ${f.}$ "

The definition above has been tailored to the notes in this volume. A more general treatment of functions is found in the set theory notes on functions. Functions are abstractions that assign (i.e., map) every element of one set to a unique element of another set. They do not have to assign numbers to numbers. We can assign apples to oranges, sets to sets, or functions to functions. Granting passports to applicants is a function, because no single passport can be granted to multiple applicants. Assigning the set of all humans to the statuses "dead" or "alive" is a function, because no human can be both dead and alive. This general construct was proposed by the German mathematician Peter Lejeune Dirichlet (1805-1859), in lieu of the old definition where functions were simply formulas. Unfortunately, many elementary schools are still stuck in the 18th century (not necessarily through a fault of their own) and continue to present functions as formulas.

Some authors make a distinction between "natural domains" and "domains." For example, the function ${f(x)=1/x}$ — couched in such authors' terms — has a domain of ${\reals}$ and a natural domain of ${\reals \smallsetminus \set{0}.}$ The distinction starts a quick path to cognitive goose chasing. We just said that functions map all the elements of their domain to a unique element of its codomain, so what's this business of some elements not getting mapped? Fortunately, the fix is simple: ${\dom{f}=\reals \smallsetminus \set{0}.}$ The notation ${\smallsetminus \set{0}}$ merely specifies the domain, which is whatever we say it is, footgun or otherwise. If not every element in the domain has a unique outcome, then we don't have a function. End of story.

identity function. We call the function ${f: X \to X,}$ or, equivalently, ${f(x)=x,}$ the identity function. Given ${X\subseteq\reals,}$ we denote the identity function on ${X}$ with the notation ${\tx{Id}_X.}$

equality of functions. We say that two functions ${f}$ and ${g}$ are equal and write ${f = g}$ if, and only if ${\dom{f}=\dom{g}}$ and ${\ran{f}=\ran{g}.}$

algebra of functions. Given the real functions ${f:X\mapsto{Y}}$ and ${g:A\mapsto{B}}$ with ${X \cap A \neq \nil,}$ the following operations are defined for all ${x \in X}$ and for all ${a \in A.}$
$\eqs{ f + g &= f(x) + g(a). \\ -f &= -f(x). \\ f - g &= f(x) - g(a). \\ fg &= f(x) \by g(a). \\ \dfrac{f}{g} &= \dfrac{f(x)}{g(a)} ~\nc~ \forall a[g(a) \neq 0]. }$

Function Types

Below is a quick overview of the different function types. The function types can also be viewed from a procedural perspective. Suppose ${f:A \mapsto B}$ where ${b \in B.}$ Saying that ${f}$ is injective is akin to: "There is at most one way to reach the result ${b.}$ " When we say that ${f}$ is surjective: "There is at least one way to reach the result ${b.}$ And bijective: "There is exactly one way to reach the result ${b.}$

	${f: x \in \X \mapsto y \in \Y}$	notation	leeway
injective	${y}$ mapped to at most once	${f: \X \inj \Y}$	some ${y}$ are not mapped to
surjective	${y}$ mapped to at least once	${f: \X \surj \Y}$	some ${y}$ are mapped to more than once (but not by the same ${x}$ )
bijective	${y}$ mapped to exactly once	${f: \X \bij \Y}$	none; unique ${x}$ to unqiue ${y}$

Function Composition

function composition. Let ${f}$ and ${g}$ be functions. The composition of ${f}$ and ${g,}$ denoted ${f \wc g,}$ is defined as ${(f \wc g)(x) = f(g(x)),}$ where ${\dom{f \wc g} = \set{x \in \dom{g} : g(x) \in \dom{f}}.}$

The idea behind function composition: We take the outcome of some function (say, ${g}$ ), and pass it as an argument to another function ( ${f(g(x))}$ ).

Inverse Functions

inverse function. Given a function ${f: X \inj Y,}$ we call the relation ${f^{-1}: Y \inj X}$ the inverse of ${f.}$

Informally: Given a function ${f:X \mapsto Y,}$ we call the function ${f^{-1}: Y \mapsto X}$ the inverse of ${f.}$ In other words, if ${f(x)=y}$ then ${f^{-1}(y)=x.}$ We must be careful with the notion of an inverse function, because the inverse of a function ${f}$ is not necessarily a function. Suppose ${A=\set{1,2,3},}$ ${B=\set{a,b,c,d},}$ and ${f = \set{(1,a), (2,b), (3,c)}.}$ Given that ${f}$ is an injective function, we have ${f^{-1}=\set{(a,1),(b,2),(c,3)}.}$ But ${f^{-1}}$ is not a function, since the element ${d}$ doesn't have a unique outcome. Accordingly, we refrain from the characterization of inverses as "undoing" a function, because that's not really what's happening here (and in this author's experience, it leads to a tempting thought that the inverse is a function). As such, we treat a function's inverse as an entirely different construct. It might be a function or not a function at all.

Curves

curves. Given a function ${f,}$ we call the set ${\Gamma_f = \set{(x,y)\in \reals^2:y=f(x)}}$ the curve of ${f}$ on the plane ${\reals^2.}$

Branching Functions

Some functions are conditional, and in that they apply different procedures depending on the arguments passed. We call such functions branching functions (or piecewise functions). The most common example is the absolute value function:

\abs{x} = \case{ x &\if x \ge 0 \\ -x &\if x \lt 0 }

Equality

The absolute value function allows us to define the notion of equality for the reals.

definition. Let ${a, b \in \reals.}$ We say that ${a}$ is equal to ${b}$ and write ${a = b}$ if, and only if, for every real number ${\varepsilon \gt 0,}$ it follows that ${\abs{a - b} \lt \varepsilon.}$

Classical Inequalities

triangle inequality. ${\forall(a,b) \in \reals^2:\abs{a+b} \le \abs{a} + \abs{b}.}$

corollary. ${\forall x_1, \ldots, x_n \in \reals:(x_1 + \ldots + x_n) \le (\abs{x_1} + \ldots + \abs{x_n}).}$

corollary. ${\forall(a,b) \in \reals^2~:~\abs{a-b} \ge \abs{\abs{a}-\abs{b}}.}$

Intervals

definition. An interval is a set ${I \subset \reals}$ such that, for all ${x,y,z \in I,}$ if ${x \lt y \lt z}$ and ${x,z \in I,}$ then ${y \in I.}$ Given ${a,b \in \reals,}$ we define the following: The interval ${\set{x \in \reals:a \lt x \lt b}}$ is called a closed interval. The interval ${x\in\reals: a \le x \le b}$ is called an open interval. The interval ${\set{x\in\reals: a \lt x \le b}}$ is called a right-open interval. The interval ${\set{x\in\reals:a \le x \lt b}}$ is called a left-open interval.

We will always denote intervals with sets. We will not use notations like ${(a,b),}$ ${[a,b)}$ or ${]a,b[,}$ since the sets aren't all that difficult to write and they're clearer.

Properties of Valued Functions

Monotonicity

definition. Given ${I \subset \reals,}$ and ${R \subseteq \reals,}$ and ${f: I \mapsto R,}$ the following properties are defined:

${f}$ is increasing if, and only if, for all ${a,b \in I,}$ if ${a \lt b,}$ then ${f(a) \le f(b).}$

${f}$ is decreasing if, and only if, for all ${a,b \in I,}$ if ${a \lt b,}$ then ${f(a) \ge f(b).}$

${f}$ is strictly increasing if, and only if, for all ${a,b \in I,}$ if ${a \lt b,}$ then ${f(a) \lt f(b).}$

${f}$ is strictly decreasing if, and only if, for all ${a,b \in I,}$ if ${a \lt b,}$ then ${f(a) \gt f(b).}$

${f}$ is monotonic if ${f}$ is increasing or decreasing.

${f}$ is strictly monotonic if ${f}$ is strictly increasing or strictly decreasing.

Symmetry

definition. Given a funtion ${f}$ and a number ${x \in \dom{f} \subseteq \reals,}$ we define the following:

${f}$ is an even function iff for all ${x \in \dom{f},}$ ${f(-x) = f(x).}$

${f}$ is an odd function iff for all ${x \in \dom{f},}$ ${f(-x)=-f(x).}$

The descriptors "even" and "odd" come from the powers of some ${(-x).}$ Given ${x^n}$ where ${n,k \in \nat,}$ if ${n=2k}$ (an even number) then ${(-x)^{n}=x.}$ If ${n=2k+1,}$ then ${(-x)^n=-x.}$ Even functions have the special property of being symmetric about the ${y}$ -axis, since all their tuples are either ${(-x,y)}$ or ${(x,y).}$ Odd functions have the property of being symmetric about the origin: ${(-x,-y)}$ or ${(x,y).}$