# What Is a Function?

*All things are subject to interpretation whichever interpretation prevails at a given time is a function of power and not truth.*(Friedrich Nietzsche)*The function of muscle is to pull and not to push, except in the case of the genitals and the tongue.*(Leonardo da Vinci)*The supreme function of reason is to show man that some things are beyond reason. (*Blaise Pascal)*The function of education is to teach one to think intensively and to think critically…Intelligence plus character – that is the goal of true education.*(Martin Luther King, Jr.)

# Functions

# Domain

We denote by the set of rational numbers, that is, the numbers with finite or periodic decimal expansion. We write if for each given value of in the *domain* of , a rational number is assigned, that is, for each the function value is assigned.

# Range

The set of values for forms the *range* of . We can thus write stating that for each there is assigned a value , and for each value there is at least one value such that .

# f: From Domain to Range

We write to indicate that for each a specific value is assigned in the sense that .In particular, writing , means that for each , a function value is assigned.

Writing means that for each -vector , an -vector with rational coefficients is assigned.

If is defined for in a subset of , which we write expressing that if then , then we write .

We often write e.g. without explicitly specifying the domain , or the range .

It is common to denote by the set of all rational numbers together with the numbers with an infinite non-periodic decimal expansion, referred to as the set of \emph{real numbers}.

We shall see a function $f: \mathcal{Q}^m\rightarrow \mathcal{Q}^n$ can uniquely be extended to a function $f: \mathcal{R}^m\rightarrow \mathcal{R}^n$, if the function is *continuous* in a way to be specified precisely below (*Lipschitz continuous*).

We denote by the set of positive real numbers.

# Read More

- \hyperref[whatisafunction]{What is a Function?}

# To Think About

- How did the notion of
*function*develop? - Who introduced the concept and terminology

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