I'm not sure how to go about proving this theorem:
Let $U\subset \mathbb{R}^m$ (open set) and $f:U\longrightarrow \mathbb{R}^n$ a differentiable function such that:
$\forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,[x,x+h]\subset U \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!|<\epsilon |\!|h|\!|$
then it holds that $f':U\longrightarrow \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ is continuous.We can also say that $f'$ is uniformly continuous?
Any hints would be appreciated.