Epsilon-Function Differentiability Condition

Theorem
Let $\mathbb K$ be either $\R$ or $\C$.

Let $f: D \to \mathbb K$ be a continuous mapping, where $D \subseteq \mathbb K$ is an open set.

Let $z \in \mathbb K$.

Then $f$ is differentiable at $z$ there exist $\alpha \in \mathbb K$ and $r \in \R_{>0}$ such that for all $h \in \map {B_r} 0 \setminus \set 0$:


 * $\map f {z + h} = \map f z + h \paren {\alpha + \map \epsilon h}$

where $\map {B_r} 0$ denotes an open ball of $0$, and $\epsilon: \map {B_r} 0 \setminus \set 0 \to \mathbb K$ is a continuous mapping with $\displaystyle \lim_{h \mathop \to 0} \map \epsilon h = 0$.

If the conditions are true, then $\alpha = \map {f'} z$.

Also see

 * Characterization of Differentiability