# When does convergence of function imply convergence of its derivative?

Let $$F_n$$ be a sequence of differentiable real valued functions.

Suppose that $$\lim_{n \to \infty} F_n(x) = F(x)$$ and that $$F(x)$$ is differentiable.

Under which conditions does that imply

$$\lim_{n \to \infty} F'_n(x) = F'(x)$$?

Do I need some regularity, or maybe that the $$F_n$$ converges uniformly?

You need to add the assumption that $F_n'$ converges uniformly on a closed interval $[a,b]$. In fact:
Theorem: Suppose $\{f_n\}$ is a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point $x_0$ on $[a,b]$. If $\{f_n'\}$ converges uniformly on $[a,b]$, then $\{f_n\}$ converges uniformly on $[a,b]$, to a function $f$, and $$f'(x)=\lim_{n\to\infty}f_n'(x),\quad(a\leq x\leq b).$$
• thank you! But doesn't that give the converse implication? I mean you assume that the $F_n'$ converge uniformly and conclude that also $F_n$ converge uniformly (if they converge in a single point, which I think it's just a way to make sure that the indefinite constant are the same). I was interested in the other implication – Ant Mar 17 '15 at 17:32