Q:

Does this lemma have a more common name?

Consider a function $f:(a,b)\rightarrow \mathbb{R}$ that is differentiable at $c\in (a,b)$. If $f'(c)>0,$ then there exists $\epsilon>0$ so that $f:(c-\epsilon,c+\epsilon)\rightarrow \mathbb{R}$ is strictly increasing. If $f'(c)<0$, then there exists $\epsilon>0$ so that $f:(c-\epsilon,c+\epsilon)\rightarrow \mathbb{R}$ is strictly decreasing.
Is there a common name for this lemma? I am trying to find an explanation.

A:

This is wrong and therefore has no name.
Let $$ f(x)=\begin{cases}x+x^2&x\in\Bbb Q\\x&x\notin \Bbb Q\end{cases}$$
Then $f'(0)=1$, but $f$ is not strictly increasing on any open interval.

