Function (map)

A function (or map) from a set $X$ to a set $Y$ is a rule that assigns to each $x\in X$ exactly one element of $Y$, denoted $f(x)$. Formally, it is a subset $f\subseteq X\times Y$ (a Cartesian product ) such that:

• for every $x\in X$ there exists $y\in Y$ with $(x,y)\in f$, and
• if $(x,y_1)\in f$ and $(x,y_2)\in f$, then $y_1=y_2$.

One writes $f:X\to Y$ and calls $X$ the domain and $Y$ the codomain . Functions are the primary objects of study in analysis: limits, continuity , differentiability , and integrability are properties of functions.

Examples:

• $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^2$.
• The projection $\pi_1:A\times B\to A$ given by $\pi_1(a,b)=a$.
• If $X=\varnothing$, there is exactly one function $f:\varnothing\to Y$ for any set $Y$ (the empty relation).