Let $f:U\rightarrow \mathbb{R}^n$ be a function, where $U\subset \mathbb{R}^m$ is open. Suppose there exist functions $f_k:U\rightarrow \mathbb{R}^n,$ $k\in\mathbb{N}$, such that for all $x\in U$: $$f(x)=\sum_{k=1}^{\infty}f_k(x).$$
Moreover, suppose that each $f_k$ is differentiable at some point $x_0\in U$. Does this imply that $f$ is differentiable at $x_0$?
Unfortunately, I do not know the answer. I checked some examples, and it seems to be correct. I would be thankful, if someone can help me.
$$f(x)=\sum_{k=1}^{\infty}f_k(x)=\lim_{n\to \infty}\sum_{k=1}^{n}f_k(x)=\lim_{n\to \infty} F_n(x),$$ for $F_n(x)=\sum_{k=1}^{n}f_k(x)$.
So if we set $f_{n+1}=h_{n+1}-F_n$ for some differenitable function $h_{n+1}$, then $F_{n+1}=F_n+f_{n+1}=h_{n+1}$.
Thus the question reduces to showing that a limit of diffenretiable functions is differentiable.
The answer is no!
Its not hard to come up with an example. Take your favorite function which is not differentiable at $0$ (e.g. |x|) and try to approximate it by differentiable functions.
Can you do that?