Function

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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. Lets call this function $f$ . A common notation to define $f$ is: $f(x) = x^2$ . 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:

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

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

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

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

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

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.

The graph of a function

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

The Inverse of a Function

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.

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 $\displaystyle |A|\leq |B|$ .

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.

Olympiad and University Level 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 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

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.

Integrability

Convexity

History of the concept

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, 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.

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