Fréchet Derivative

A function $f$ is Fréchet differentiable at $a$ if

$$\lim_{x\to a} {f(x)-f(a)\over x-a}$$

exists. This is equivalent to the statement that $\phi$ has a removable Discontinuity at $a$, where

$$\phi(x)\equiv {f(x)-f(a)\over x-a}.$$

Every function which is Fréchet differentiable is also Carathéodory differentiable.

See also Carathéodory Derivative, Derivative