Produced by Charles Wells Revised 2015-12-03 Introduction to this website website TOC website index blog Back to head of functions chapter

The examples given here:

- are basic types of functions that are used everywhere in abstract math
- are functions you need to be familiar with
- are (sigh) boring.

For any set $A$, the identity function ${{\operatorname{id}}_{A}}:A\to A$ is the function that takes each element of $A$ to itself.

- This means that for any set $A$, for every element $a\in A$, ${{\operatorname{id}}_{A}}(a)=a$. For example, ${{\operatorname{id}}_{\mathbb{Z}}}(42)=42$.
- The identity function on a set $A$ can be thought of as the function that
**does nothing**to each element of A. - The identity function on $\mathbb{R}$ is the familiar function defined by $f(x)=x$.
- Another common notation for the identity function on $A$ is "${{1}_{A}}:A\to A$."
- The word "identity" has two other commonly used meanings.
- There is a different identity
function for each different set. (See overloaded
notation.) These functions all have the “same” formula: ${{\operatorname{id}}_{A}}(a)=a$ for every $a\in A$. But they are technically
*different functions*because they have different domains.

If $A\subseteq B$ (see inclusion), then the inclusion function $\operatorname{inc}:A\to B$ that takes every element in $A$ to the same element regarded as an element in $B$. In other words, $\operatorname{inc}(a)=a$ for every element $a\in A$.

- The inclusion map is often written $A\hookrightarrow B$.
- The graph of $\text{inc}$ is the same as the graph of ${{\operatorname{id}}_{A}}$ and they have the same domain, so that the only difference between them is what the codomain is.
- For any set $A$, $A\subseteq A$, so the identity function ${{\operatorname{id}}_{A}}:A\to A$ is an inclusion function.

If $A$ and $B$ are nonempty sets and $b$ is a specific element of $B$, then then constant function ${{C}_{b}}:A\to B$ is the function that takes every element of $A$ to $b$; that is, ${{C}_{b}}(a)=b$ for all $a\in A$.

- The notation ${{C}_{b}}$ is not common. There is no standard notation for constant functions.
- A
constant function ${{C}_{b}}$
takes any element of $A$ to $b$. It has a
**one-track mind.** - A
constant function from $\mathbb{R}$ to $\mathbb{R}$ has a
**horizontal line**as its graph. - See the Wikipedia article Constant function.

If $A$ is any set, there is exactly one function $E:\varnothing \to A$. This function is called the empty function.

- An
identity function does nothing. An empty function
**has nothing to do.** - The graph of the empty function is the empty set, and so is its image.
- For any set $A$, there is exactly one empty function from $\varnothing\to A$.
- The empty function from $\varnothing\to A$ is an inclusion function vacuously.

If $A$ and $B$ are sets, there are two coordinate functions ${{p}_{1}}:A\times B\to A$ and ${{p}_{2}}:A\times B\to B$, defined for $a\in A$ and $b\in B$ by ${{p}_{1}}(a,b)=a$ and ${{p}_{2}}(a,b)=b$.

- $A\times B$ is the set of all ordered pairs $(a,b)$ with $a\in A$ and $b\in B$. See cartesian product.
- Coordinate functions are also called
**projections.** - The coordinate function ${{p}_{i}}$ may be denoted by "${{\pi }_{i}}$".
- In general for an $n$-fold cartesian product, the function ${{p}_{i}}$ takes an $n$-tuple to its $i$th coordinate.
- If either $A$ or $B$ is empty, then so is $A\times B$. In that case, each ${{p}_{i}}$ is the empty function.
- For any
set $S$, there are two
*different*coordinate functions ${{p}_{1}}:S\times S\to S$ and ${{p}_{2}}:S\times S\to S$. For example, if $S$ is the set of real numbers, then ${{p}_{1}}(2,-3)=2$ and ${{p}_{2}}(2,-3)=-3$.

A binary operation on a set $S$ is a function $F:S\times S\to S$. (See cartesian product).

- The operation
of
**adding**two real numbers gives a binary operation $(x,\,y)\mapsto x+y:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$. **Subtraction**$-:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is also a binary operation on the real numbers. Observe that, unlike addition, it cannot be regarded as a binary operation on the*positive*real numbers.**Multiplication**of real numbers is also a binary operation $(x,\,y)\mapsto xy:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$.**Division**is*not*a binary operation on the real numbers because you can’t divide by $0$. However, it is a binary operation $(x,\,y)\mapsto x/y:\mathbb{R}^*\times\mathbb{R}^*\to\mathbb{R}^*$ on the*nonzero*real numbers ("${\mathbb{R}^{*}}$" is standard notation for the set of all nonzero real numbers).- Now look at the function $(x,\,y)\mapsto
x/y:\mathbb{R}\times\mathbb{R}^*\to \mathbb{R}$. It is well defined since $0/y$ is defined even though $y/0$ is not. This is a perfectly good function but strictly speaking it is not a binary operation because by definition a binary
operation has to fit the pattern $F:S\times S\to S$ where
*all three sets are the same.* - For any set $S$, the two projections ${{p}_{1}}:S\times S\to S$ and ${{p}_{2}}:S\times S\to S$ are both binary operations on $S$.
- The
**greatest common divisor**function $\operatorname{GCD}:\mathbb{N}\times\mathbb{N}$ is a binary operation on the set of all nonnegative integers. - With a
binary operation
*symbol*, infix notation is usually used: the name of the binary operation is put*between the arguments*. For example we write $3 + 5 = 8$, not $+(3, 5) = 8$. Note that there are no symbols for the $\text{GCD}$ function and for the projections from a cartesian product. - Binary operations are the basis of most of algebra.

In this section I give you examples of really weird functions that you may never have thought of as functions before, because if you are a beginner in abstract math, you probably need to:

Loosen up narrow-minded ideas about what a function is

Other consciousness-expanding examples of functions are listed in an appendix.

A function can be
given by different rules on different parts of its domain. It is still *one
function*.

Let $F:[0,1]\to\mathbb{R}$ be defined by \[F(x):=\left\{ \begin{align} 2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(0\le x\le 0.5) \\ 3(x-1)^2\,\,\,\,(0.5\lt x\le 1) \\ \end{align}\right.\] This is the graph of $F$:

- $F$ is given by a
**split definition**. It is defined by one formula for part of its domain and by another on the rest. $F$ is nevertheless**one function**, defined on the closed interval $[0,1]$. - $F$ is
*not continuous*at $x=0.5$. - $F$ does not have a derivative at $x=0.5$
- The
graph
*does not and cannot*show the precise behavior of the function near $x=0.5$. - The point $\left( 0.5,1 \right)$ is on the graph, because the
definition of $F$ says that $F(x)=2x$ for $x$ between $0$
and $0.5$
*inclusive*. - For any point $x$ to the right of $0.5$, $F(x)=3{{\left( x-1 \right)}^{2}}$.
- $\underset{x\to {{0.5}^{-}}}{\mathop{\lim }}\,F(x)=1$ but $\underset{x\to {{0.5}^{+}}}{\mathop{\lim }}\,F(x)=0.75$.
- Nevertheless, $F(0.5)=1$, not $0.75$. That implies that $F$ is
*not continuous*at $x=0.5.$ - It would be wrong to say something like: “$F(x)=3{{\left(
x-1 \right)}^{2}}$
starting at the first point to the right of $x=0.5$”.
*There is no "first point to the right of*$0.5$." See density.

(1) A function may not be defined by a formula, and (2) it need not involves numbers at all.

Let
the function $F$ be defined on the set $\left\{\textsf{a},\textsf{b},\textsf{c},\textsf{d }\right\}$ like this: \[F(\textsf{a})=\textsf{a},\,F(\textsf{b})=\textsf{c},\,F(\textsf{c})=\textsf{c},\,F(\textsf{d})=\textsf{b}\]
In this definition, $\textsf{a}$, $\textsf{b}$, $\textsf{c}$ and $\textsf{d}$ are *letters of the alphabet,* not variables.

- $F$ is defined
*only*for inputs $\textsf{a}$, $\textsf{b}$, $\textsf{c}$ and $\textsf{d}$. For example, neither "$F(4)$" nor "$F(\textsf{u})$" are defined. - I have not defined $F$ using a formula. $F(\textsf{b})=\textsf{c}$
*entirely and only because the definition of $F$ says it is.* - You could define $F$ by saying "Let $F$ be defined on the set $\left\{\textsf{a},\textsf{b},\textsf{c},\textsf{d }\right\}$ by the diagram below." That definition gives the same function.
- You could also define the function by showing a
**table:**
\[\begin{array}{cc}
x & F(x) \\
\textsf{a} & \textsf{a} \\
\textsf{b} & \textsf{c} \\
\textsf{c} & \textsf{c} \\
\textsf{d} & \textsf{b} \\
\end{array}\]
- Databases involve many functions that don't involve numbers. For example the personnel database of a company will likely contain a function that takes the name of each employee to their job title.

The domain and codomain of a function can be completely different sets.

Let $S$ be some set
of English words, for example the set of words in a given dictionary. Then the **length
of a word** is a function; call it $L$.

- $L$ takes English words as inputs.
- $L$ outputs the number of letters in the word. For example, $L('\text{cat}')=3$ and $L('\text{polysyllabic}')=12$.
- The method of computation for this function is: count the number of letters. In a computer program, an English word would be stored as a list of characters, and most any computer language has a command that returns the length of a list.

You can correctly define a function you know you can't calculate.

A **prime pair** consists of a pair of primes that differ by $2$. The first six prime pairs are $(3,5)$, $(5,7)$, $(9,11)$, $(11,13)$, $(17,19)$ and $(29,31)$.

Let $PP:\mathbb{N}\to\mathbb{N}\times\mathbb{N}$ be defined like this: (1) If there are fewer than $n$ prime pairs, then $PP(n)=0$. (2) Otherwise, $PP(n)$ is the $n$th prime pair in order.

- For example $PP(6)=(29,31)$.
- It is not known whether there is only a finite number of prime pairs or an infinite number.
- The largest prime pair known has order of magnitude $10^{19}$. We do not know the value of $n$ for which this number is $PP(n)$.
- So for very large numbers $n$
*we know that we do not know how to calculate*$PP(n)$. Nevertheless, $PP$ is a perfectly well-defined function. - There is more information about prime pairs in the Mathworld article Twin primes and in the Wikipedia article Twin primes.

A real-valued function may not have a graph you can draw.

Let $F:\mathbb{R}\to\mathbb{R}$ be defined by \[F(x):= \begin{cases} 1 & \text{if }x\text{ is rational}\\ \frac{1}{2} & \text{if }x\text{ is irrational}\\ \end{cases}\] for all real $x$.

- $F$ is called the
**Dirichlet function.** - $F(1/3)=1$, $F(-42)=1$, but $F(\pi)=\frac{1}{2}$ because $\pi$ is not rational.
- $F$ is not continuous.
- The "graph" above is completely misleading.
*Both horizontal lines are full of holes.* - The point $(\sqrt{2},\frac{1}{2})$ is missing from the top line because $\sqrt{2}$ is not rational.
- The point $(1.2,1)$ is missing from the lower line because $1.2=\frac{12}{10}$, which
*is*rational. - This means that the left vertical line does not intersect the lower line and the right vertical line does not intersect the top line.
- The "graph" is worse than that.
- Both lines are missing
*infinitely many points.* - No matter how much you magnify either line, you will not see any of the holes, because no matter how small a nonzero distance you name -- call it $\epsilon$ -- there is a point that
*is*on the line that is closer than $\epsilon$ to the hole. - There is also a hole closer than $\epsilon$ to any point that
*is*on the line. - See Density of the rationals and Density of the reals.
- You can read more about this function in the Wikipedia article Dirichlet function.

The graph of a real valued function on a finite interval can be an infinitely long curve.

Let $F(x):=.5 \sin \left( \frac{50}{x} \right),\,\,\,\,\,\,\left( x \gt0 \right)$.

- The frequency of the curve goes up rapidly as you get close to the $y$-axis from the right, since $\frac{50}{x}$ grows very rapidly as $x$ moves toward $0$.
- Drawing the graph near the $y$-axis is impossible because the
curve between $x = 0$ and any larger number is
*infinitely long*even though it occurs in a finite interval. - More about this function.

A function can have a function as input and a function as output.

Let $\mathcal{D}$ be the set of all differentiable functions from $\mathbb{R}$ to $\mathbb{R}$, and $\mathcal{A}$ the set of all functions from $\mathbb{R}$ to $\mathbb{R}$. Define $D:\mathcal{D}\to\mathcal{A}$ be the function that takes a function $f$ in $\mathcal{D}$ (in other words a function that has a derivative) to its derivative.

- If $f(x):={{x}^{2}}$ then $\text{D}(f)(x)=2x$. Using barred arrow notation, $D(x\mapsto {{x}^{2}})=x\mapsto 2x$.
- If $f(x):=\sin x$ then $\text{D}(f)(x)$ is $x\mapsto\cos x$, in other word $D(x\mapsto\sin x)=x\mapsto\cos x$.
- D takes a function as input and outputs another function,
namely the derivative of the first one. The
*whole function is the input*, not some value of the function, not the rule that defines the function, not the graph. You have to think of the function as a*single thing*, in other words as a math object. - Functions whose inputs are
complicated structures such as functions may be called operators
*.**(Usage varies in different specialties.)***differentiation operator**. - The differentiation operator is not injective. For example, $f(x):={{x}^{2}}$ and $g(x):={{x}^{2}}+42$ have the same derivative, namely $2x$.

The chapter Derivatives contains many graphs, each showing a function and several of its derivatives.

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