Function

A function is a rule that maps one set of values to another set of values. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value, and squares it to get an output value. One can call this function $f$ .

Rigorous Definition

Let $A$ , $B$ be sets

Let $f\subset A\times B$ , ( $A\times B$ is the Cartesian Product of $A$ and $B$ )

We say that $f$ is a function from $A$ to $B$ if and only if

[ $(a,b)\in f$ $\wedge$ $(a,c)\in f$ ] $\implies$ $b=c$ ( $(a,b)$ is the notation for an Ordered Pair)

Introductory Topics

Domain and Range

The domain of a function is the set of input values for the argument of a function. The range of a function is the set of output values for that function. For an example, consider the function: $f(x) = \sqrt{x^2-9}$ . The domain of the function is the set ${x:|x|>3}$ , where $x$ is a real number. The range is the set of all non-negative real numbers because the square root can never return a negative value.

Real Functions

A real function is a function whose range is in the real numbers. Usually we speak about function whose domain is also a subset of the real numbers.

Graphs

Functions are often graphed. To find out if a graph is a function, it must stand up to the vertical line test.

Inverses

The inverse of a function is a function that "undoes" a function. For an example, consider the function: f(x) $= x^2 + 6$ . The function $g(x) = \sqrt{x-6}$ has the property that $f(g(x)) = x$ . In this case, $g$ is called the (right) inverse function. (Similarly, a function $g$ so that $g(f(x))=x$ is called the left inverse function. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.) Often the inverse of a function $f$ is denoted by $f^{-1}$ .

Intermediate Topics

Injections, Surjections, Bijections

An injection (or one-to-one function) is a function which has distinct values for distinct arguments within a given domain.

By definition, $f:A\to B$ is injective if $f(a)=f(b) \Rightarrow a=b$ , or equivalently, $a\neq b \Rightarrow f(a)\neq f(b).$

Examples:

$f(x) = x$ is injective from $\mathbb{C} \rightarrow \mathbb{C}.$
$f(x) = x^2$ is injective from $\mathbb{R^+} \rightarrow \mathbb{R}.$
$f(x) = x^2$ is not injective from $\mathbb{R} \rightarrow \mathbb{R}.$

Injectivity of a function $f: A \rightarrow B$ implies that $f$ has an inverse. Furthermore, if $A$ and $B$ are finite sets, injectivity implies $|A|\leq |B|$ .

A surjection (or onto function) maps at least one element from its domain, $A,$ onto every element of its range, $B.$

Examples:

$f(x) = x$ is surjective from $\mathbb{C} \rightarrow \mathbb{C}.$
$f(x) = x^2$ is surjective from $\mathbb{R} \rightarrow \mathbb{R^+}$
$f(x) = x^2$ is not surjective from $\mathbb{R} \rightarrow \mathbb{R}.$

A bijection (both one-to-one and onto) is a function, $f: A \rightarrow B$ that is both injective and surjective.

If $f$ is an injection from $A \rightarrow B$ and $g$ is an injection from $B \rightarrow A,$ then there exists a bijection, $h,$ between $A$ and $B$ by the Schroder-Bernstein Theorem.

Monotonic functions

A function $f:A\to B$ is called monotonically increasing if $f(x_1)\geq f(x_2)$ holds whenever $x_1>x_2$ . If the inequality holds strictly $(f(x_1)>f(x_2))$ , then the function is called strictly increasing.

Similarly, a function $f:A\to B$ is called monotonically decreasing if $f(x_1)\geq f(x_2)$ holds whenever $x_1<x_2$ . If the inequality holds strictly $(f(x_1)>f(x_2))$ , then the function is called strictly decreasing.

Advanced Topics

Functions of Real Variables

A real function is a function whose range is in the real numbers. Usually we speak about function whose domain is also a subset of the real numbers.

Continuity

Intuitively, a continuous function has the propriety that its graph can be drawn without taking the pencil off the paper. But the reality about continuous functions is more complex.

Epsilon-Delta Definition

A function $f:E\to\mathbb{R}$ is called continuous at $x_{0}$ if, for all $\varepsilon >0$ , there exists $\delta >0$ such that $|x-x_0|<\delta$ and $x\in E \Rightarrow |f(x)-f(x_0)|<\varepsilon$ .

Heine Definition

The previous definition of continuity at $x_{0}$ is equivalent with the following: for every sequence $(x_n)_{n\geq 0}$ such that $\lim_{n\to\infty}x_n=x_0$ , we have that $\lim_{n\to\infty}f(x_n)=f(x_0)$ .

It is easy to see that a function is continuous in isolated points, and is continuous in accumulation points iff the limit of the function in those points equals the value of the function.

A function is continuous on a set if it is continuous in every point of the set.

Properties of Continuous Functions

The sum and product of two continuous functions are continuous functions.
The composition of two continuous functions is a continuous function.
In any closed interval $[a, b]$ , there exist real numbers $c$ and $d$ such that $f$ has a maximum value at $c$ and $f$ has a minimum value at $d$ .
Intermediate Value Theorem (see below)

Intermediate value theorem

If a function is continuous, then it has the Intermediate Value Theorem. The converse is not always true. Proof:...

Differentiability

For functions of one variable, differentiablility is simply the question of whether or not a derivative exists. For functions of more than one variable, it's significantly more complicated. In the case of both one and multivariable functions, differentiability implies continuity.

A single-variable function $f(x)$ is differentiable at $x=a$ iff:

$f(a)\in\mathbb{R}$
$\lim_{x\to a}f(x)\in\mathbb{R}$
$f(a)=\lim_{x\to a}f(x)$
$\lim_{x\rightarrow a} \frac{d}{dx} \in \mathbb{R}$

Integrability

All continuous functions are integrable, as well as many non-continuous functions.

Convexity

A twice-differentiable function $f(x)$ is concave up (or convex) in the interval $[a,b]$ iff $f''(x)>0$ in the interval $[a,b]$ and concave down (or concave) iff $f''(x)<0$ . The points of inflection, when the concavity switches, of the function occur at the roots of $f''(x)$ .

Notation

A common notation to define $f$ is: $f(x) = x^2$ (where the $x^2$ , of course, is merely an example). This tells us that $f$ is a function that squares its argument (its input value). Note that this "rule" can be arbitrarily complicated and doesn't need to be given by a simple formula or description. The only requirement is that $f(x)$ should be uniquely determined by $x$ . The following are examples of functions:

$f(x)=x ^ {2}+2x-2$

$f(x)=\sin(\log{x})$

$f(x)=x^2$ for $x>0$ , otherwise $f(x)= \sin{x}$

$f(x)=p(g(x))$

$g(x)=F'(x)$

History of Functions

Without being used explicitly, the notion of function first appears with the ancient Greeks and Egyptians.

The rigorous definition was stated in the 19th century and is the result of the works of some famous mathematicians: A.L. Cauchy, Leonhard Euler, and Bernhard Riemann. With the development of set theory, a new branch of mathematics appeared, mathematical analysis, in which the notion of function has a central role.

The current notation used is attributed to Leonhard Euler.