Function with $f'>0$ but not on a $\delta$ neighbourhood

Question

I am looking to find the following:

A function $f$ such that $f'$ exists on an open interval and $f'(c) >0$ for $c$ in this interval but such that there is no $\delta$-nbhood $V_\delta (c)$ of $c$ with $f'(x) > 0$ for $x$ in $V_\delta (c)$.

My idea: $$g_2(x) = \begin{cases} x^2 \sin ({1 \over x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$

But for this function $f'(0) = 0$.

Is there a way to fix this?

Answer 1

Consider $g_2(x)+a\,x$ for appropriate values of $a$.