Differentiability

Definition

Functions of the form $f:A\subseteq\mathbb{R}\rightarrow\mathbb{R}$

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

• $$f'(a)=\lim_{t\rightarrow 0}\left(\frac{f(a+t)-f(a)}{t}\right)$$ , provided this limit exists. 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 $t$ is sufficiently small that $a+t$ stays in an open set about $a$ of course.

  (Note that $\vert\cdot\vert$ corresponds to the absolute value as a metric)

To-do

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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

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