# Differentiability

## Definition

### Functions of the form [ilmath]f:A\subseteq\mathbb{R}\rightarrow\mathbb{R} [/ilmath]

Let [ilmath]f:A\rightarrow\mathbb{R} [/ilmath] be a function and suppose that [ilmath]A[/ilmath] contains a neighbourhood of the point [ilmath]a\in A[/ilmath][Note 1] We define the derivative at [ilmath]a[/ilmath] as follows:

• $f'(a)=\lim_{t\rightarrow 0}\left(\frac{f(a+t)-f(a)}{t}\right)$, provided this limit exists[1]. This is the same as:
• $\forall t\ \forall\epsilon>0\ \exists\delta>0\left[0<\vert t\vert <\delta\implies\left\vert\frac{f(a+t)-f(a)}{t}\right\vert<\epsilon\right]$ where [ilmath]t[/ilmath] is sufficiently small that [ilmath]a+t[/ilmath] stays in an open set about [ilmath]a[/ilmath] of course.
(Note that [ilmath]\vert\cdot\vert[/ilmath] corresponds to the absolute value as a metric)

## Notes

1. We a neighbourhood, [ilmath]N[/ilmath], of a point [ilmath]A[/ilmath] to mean [ilmath]\exists U\text{ that is open }[a\in U\wedge U\subseteq N][/ilmath]. In the case of a metric space our neighbourhood must contain an open ball about [ilmath]a[/ilmath]

## References

1. Analysis on Manifolds - James R. Munkres

## To-do

TODO: Right now this links to a generic limit page, I need to cover different limits (and cover somewhere that differentiability requires a normed space