Function

The notion of a function is one of the basic notions of mathematics. To define a function, you need to know its domain (the set of admissible arguments) $\displaystyle{X}$ , its codomain (the set of possible values) $Y$ , and a "rule" $\displaystyle{f}$ that assigns to every element $x\in X$ a unique element $y=f(x)\in Y$ .

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:

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

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

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

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

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

When people talk of functions, most of the time they mean functions whose domain and codomain are sets of numbers (real or complex) or vectors. Since arbitrary functions can be arbitrarily bad and hard to handle, certain "good" classes of functions have been introduced like continuous functions, differentiable functions, measurable functions, etc. The functions in these classes possess many nice properties general functions don't have.

Ways to define a function

Injections, surjections, bijections

An injective function(or one-to-one) is a function which has distinct values for distinct arguments.

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

If $\displaystyle A$ and $\displaystyle B$ are finite sets injectivity implies $\displaystyle |A|\leq |B|$ .

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.

Monotonic functions

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

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

The graph of a function

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

Continuity

Intuitively, a continuous function has the propriety that it's graph can be drawn without taking the pencil off the paper. But the reality about continuous function 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 $\displaystyle \lim_{n\to\infty}x_n=x_0$ 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 point 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 is a continuous function.
The composition of two continuous functions is a continuous function.
...

Intermediate value property

If a function is continuous then it has the Intermediate value property. The converse is not always true. Proof:...

Continuity on compact intervals

Differentiability

Integrability

Convexity

History of the concept

Without being used explicitly, the notion of function first appears at the ancient greeks and egyptians.

The rigurous definition was stated in the 19th century and is the result of the works of some famous mathematicians: A.L. Cauchy, L. Euler, B. 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.

Page

Toolbox

Search