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:

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

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

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

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

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

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

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

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

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

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 XIXth 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 appeard, Mathematical Analysis, in which the notion of function has a central role.

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