Exchange of the one-sided derivative and the one-sided limit of a real variable Let $f$ be a real-valued continuous function on $[0, 1)$. Suppose that $f$ is infinitely-differentiable on $(0 , 1)$. Is there some sufficient condition for $f$ to be right-differentiable at $0$ with
$$f_{+}'(0) = \lim_{x \to 0^{+}} f'(x),$$
where I use the notation $f_{+}'(0) := \lim_{\Delta x \to 0^+} \frac{f(\Delta x) - f(0)}{\Delta x}$ for the right derivative of $f$ at $0$?

The existence of the limit $\lim_{x \to 0^{+}} f'(x)$ is sufficient for $f^\prime_+(0)$ to exist and in that case we have the equality
$$f_{+}'(0) = \lim_{x \to 0^{+}} f'(x).$$
You can perform the proof by applying the Mean Value Theorem to $g_a(x) = f(x) -ax$ where $a = \lim_{x \to 0^{+}} f'(x)$.
