Carathéodory Derivative

A function $f$ is Carathéodory differentiable at $a$ if there exists a function $\phi$ which is Continuous at $a$ such that

$$f(x)-f(a)=\phi(x)(x-a).$$

Every function which is Carathéodory differentiable is also Fréchet Differentiable.

See also Derivative, Fréchet Derivative