Let $f:\mathbb R\to\mathbb R$ be differentiable on $[0,\infty)$. Suppose that $\lim_{x\to0^+}f'(x)$ exists and is finite, and that $f'$ is continuous at $0$. Must there exist some $\epsilon>0$ such that $f'$ is continuous on $[0,\epsilon)$?
My hunch is that this is false.
My idea is that locally near $x=1/n$, any positive integer $n$, the function $f$ is a copy of $\frac{x^2}{n}\sin(1/x)$. I guess strictly speaking, near $x=1/n$, $f$ is locally given by $$\frac{(x-1/n)^2}{n}\sin\left(\frac{1}{x-1/n}\right).$$ Then $f'$ is discontinuous at $x=1/n$, but we should still have $f'\to0$, as the oscillations are decreasing in amplitude.
Could anyone confirm that this works and/or find a simpler counterexample?
