## Functions

### Meta

${\mathfrak D}_\mathbb{\to}$

#### Discussion

##### Many definitions are implcit

Caution: Many function definitions are implicit, sometimes secretly so. E.g. defining

$f(x):=\sum_{n=0}^\infty\frac{(-1)^{3n}}{n!}z^n$

is defining

$f(x):=\lim_{m\to\infty}\sum_{n=0}^m\frac{(-1)^{3n}}{n!}z^n$

and a Limit definition is always a Task to find said limit.

##### Practically speaking, uor functions are partitially defined

Caution: In the function definition, the domain of the function, as well as variables in context, are stated quite broadly - it must be checked if the to-be-function would return a value for any given input.

In that sense, “Functions” really is “Partial functions”, a priori, although I try to state good valid domains on AoC.

##### todo

Is there any reason to generalize $\to$ to $\prod$ here? I.e. going from $A\to{B}$ (function type) to $\prod_{a:A}B(a)$ (dependent product type).