Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $c \in \mathbb{R}$ be a critical point of f. Let f be differentiable on an open interval containing c, except possibly at c.
Can we say that whenever such a function has an isolated (strict) extremum at c, say a maximum, then $\exists \delta \gt 0$ such that $f'(x) \gt 0; \ \forall x \in \left( c-\delta, c\right)$ and $f'(x)\lt 0; \ \forall x \in (c,c+\delta)$, basically can we say if f has an isolated extremum at c then the sign of f' changes at c? If so, can someone please provide a hint to the proof?