Function

A function [ilmath]f[/ilmath] is a special kind of relation

Domain

A function ought be defined for everything in its domain, that's for every point in the domain the function maps the point to something.

(See notation below if you're not sure what the [math]f:X\rightarrow Y[/math] notation means)

[math]f:\mathbb{R}\rightarrow\mathbb{R}[/math] given by [math]f(x)=\frac{1}{x}[/math] isn't defined at [math]0[/math]
[math]f:\mathbb{R}\rightarrow\mathbb{R}[/math] given by [math]f(x)=x^2[/math] is correct, it is not surjective though, because nothing maps onto the negative numbers, however [math]f:\mathbb{R}\rightarrow\mathbb{R}_{\ge 0}[/math] with [math]f(x)=x^2[/math] is a surjection. It is not an injective function as only [math]0[/math] maps to one point.

A function [ilmath]f[/ilmath] from a domain [ilmath]X[/ilmath] to a set [ilmath]Y[/ilmath] is denoted [ilmath]f:X\rightarrow Y[/ilmath]
If we have say two topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] then we may write:
- [ilmath]f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})[/ilmath] and mean [ilmath]f:X\rightarrow Y[/ilmath]
That is to say that as a general rule given a function [ilmath]f:(A_1,A_2,\cdots)\rightarrow(B_1,B_2,\cdots)[/ilmath] take it as a function [ilmath]f:A_1\rightarrow B_1[/ilmath]
- A tuple makes no sense there anyway, for multiple arguments we use the Cartesian product anyway.

TODO: Come back after the relation page and fill this out