TITLE: Sufficient condition to show $f$ is monotonically increasing in some neighborhood
QUESTION [5 upvotes]: I am curious if the following statement holds. 
Let $f:[a,b] \rightarrow \mathbb{R}$ be a continuous function differentiable on the open interval $(a,b)$. Then if $f'(c)>0$ for some $c \in (a,b)$, there exists a neighbourhood of $c$ in which $f$ is monotonically increasing. 
An ideal answer to this question would include either a proof or a counterexample.

REPLY [6 votes]: As noted in other answers the results holds if $f\in C^1$. Consider the following counterexample:
$$f(x)=\begin{cases}
x+2x^2 \sin(\frac{1}{x})\quad \text{ if } x\neq 0 \\
0\qquad \quad \qquad \quad \quad \text{ if } x=0
\end{cases}$$
with derivative
$$f^\prime(x)=\begin{cases}
1+4x \sin(\frac{1}{x})-2\cos(\frac{1}{x}) \quad \text{ if } x\neq 0 \\
1\quad \qquad\qquad\qquad\quad \quad\qquad  \text{ if } x=0
\end{cases}$$
Note that $f^\prime(0)>0$ but every neighbourhood of $0$ has negative values (and positive values).