Originally Posted by

**bram kierkels** I say $\displaystyle f$ is *differentiable at $\displaystyle x_0$ with derivative $\displaystyle L$* if we have

$\displaystyle \lim_{x\rightarrow x_0,x\in E-{x_0}}\frac{||f(x)-(f(x_0)+L(x-x_0))||}{||x-x_0||}=0

$

where $\displaystyle ||x||$ is the length of $\displaystyle x$ as measured in the $\displaystyle l^2$ metric

I say $\displaystyle f$ is *differentiable in the direction $\displaystyle v$ at $\displaystyle x_0$* if the limit

$\displaystyle \lim_{t\rightarrow 0;t>o,x_0+tv\in E}\frac{f(x_0+tv)-f(x_0)}{t}$ exists.