Let $f : [0, 1] \rightarrow \mathbb{R}$ be continuous. Suppose that $f $ is differentiable on $(0, 1)$ and $\lim_{x \rightarrow 0} f^{'}(x)=\alpha$ for some $\alpha\in \mathbb{R}$. Show that $f^{'}(0)$ exists and $f^{'}(0) = \alpha$.
How should I approach this problem. I need some hints. Second thing, are we trying to prove here that $f^{'}$ is continuous at $0$. We know $f$ is not differentiable at $0$ because $f$ is not even defined in the neighbourhood of $0$.
What is the intuition behind this question? I just do not want to solve it, I want to see the background process.