Q) Consider a function $f:(0,1) \to \mathbb{R}$, continuous annd differentiable. $(\alpha_n)^\infty_1$ be a sequence of roots of $f$ which converges to $\alpha \in (0,1)$. Prove that $f'(\alpha)=0$.
Here I am using Rolle's Theorem get a sequence $(c_n)_1^\infty$ such that $f'(c_n)=0$ and $Lt \, c_n=\alpha$. But after this I am not being able to prove that $f'(x)$ is continuous so that I can use the sequential criteria.