Let $f$ be a function differentiable in $x_0 \in \mathbb{R}$ (and its neighborhood) and $f'(x_0)>0$. Does this imply that there exists a $\delta>0$ for which $f$ is increasing in $(x_0-\delta, x_0+\delta)$?
My intuition is that since the condition is $f'(x_0)>0$ and not $f'(x_0)\ge0$, there is a neighborhood in which the derivative stays positive, and the answer to the question would be “yes”. Is this correct? If so, how could I write it down more formally?