Category:Epsilon-Function Differentiability Condition

This category contains pages concerning Epsilon-Function Differentiability Condition:

Let $f: D \to \R$ be a continuous function, where $D \subseteq \R$ is an open set.

Let $x \in \R$.

Then $f$ is differentiable at $x$ if and only if there exist $\alpha \in \R$ and $r \in \R_{>0}$ such that for all $h \in \openint {-r} r \setminus \set 0$:

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

where:

$\openint {-r} r$ denotes an open interval of $0$

$\epsilon: \openint {-r} r \setminus \set 0 \to \R$ is a real function with $\ds \lim_{h \mathop \to 0} \map \epsilon h = 0$.

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

If $f$ is continuously differentiable at $x$, then $\epsilon$ is a continous function.