# 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.)

# Domain

We denote by $\mathcal{Q}$ the set of rational numbers, that is, the numbers with finite or periodic decimal expansion. We write $f: D(f)\rightarrow \mathcal{Q}$ if for each given value of $x$ in the domain $D(f)$ of $f$, a rational number $f(x)$ is assigned, that is, for each $x\in D(f)$ the function value $f(x)\in \mathcal{Q}$ is assigned.

# Range

The set of values $f(x)$ for $x\in D(f)$ forms the range $R(f)$ of $f$. We can thus write $f:D(f)\rightarrow R(f)$ stating that for each $x\in D(f)$ there is assigned a value $f(x)\in R(f)$, and for each value $y\in R(f)$ there is at least one value $x\in D(f)$ such that $y=f(x)$.

# f: From Domain to Range

We write $f:D(f)\rightarrow \mathcal{Q}$ to indicate that for each $x\in D(f)$ a specific value $f(x)\in \mathcal{Q}$ is assigned in the sense that $x\rightarrow f(x)$.In particular, writing $f:\mathcal{Q}\rightarrow \mathcal{Q}$, means that for each $x\in \mathcal{Q}$, a function value $f(x)\in \mathcal{Q}$ is assigned.

Writing $f:\mathcal{Q}^m\rightarrow \mathcal{Q}^n$ means that for each $m$-vector $x\in \mathcal{Q}^m$, an $n$-vector $f(x)$ with rational coefficients is assigned.

If $f(x)\in \mathcal{Q}^n$ is defined for $x$ in a subset $\Omega$ of $\mathcal{Q}^m$, which we write $\Omega\subset \mathcal{Q}^m$ expressing that if $x\in\Omega$ then $x\in \mathcal{Q}^m$, then we write $f:\Omega\rightarrow \mathcal{Q}^n$.

We often write e.g. $f: \mathcal{Q}^m\rightarrow \mathcal{Q}^n$ without explicitly specifying the domain $D(f)\subset \mathcal{Q}^m$, or the range $R(f)\subset \mathcal{Q}^n$.

It is common to denote by $\mathcal{R}$ 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 $f(x)$ is continuous in a way to be specified precisely below (Lipschitz continuous).

We denote by $\mathcal{R}_+$ the set of positive real numbers.