The question actually came to me from series of functions. Suppose $f_n(x)$ is a sequence of continuous differentiable functions, that $$\sum_{n=1}^\infty f_n'(x)$$ converges uniformly on any closed subinterval $[a,b]\subseteq \mathbb{R}$, and that $\sum_{n=1}^\infty f_n(x)$ converges pointwise for any $x$.
Then necessarily on any subinterval the sum of $f_n(x)$ converges to a function $f(x)$ such that the sum of $f_n'(x)$ converges to $f'(x)$, and it seems somewhat intuitive to go from this to saying that the sum of $f_n(x)$ converges to a differentiable limit function on all of $\mathbb{R}$ as differentiation is a pointwise property.
But I'm still not sure about it.. I wasn't really able to prove it myself, and I'm not sure that the function I get for one interval will even match the function I get on another.
So is the claim true? Or is there an obvious counter-example?
What you have to show is that for any given $x \in \mathbb{R}$ the sequence converges to a differentiable function the derivative of which is given by the series involving the derivatives. But since any $x$ is contained in a suitably chosen compact interval (e.g. $[-|x|-1, |x| + 1$) where your assumption applies, this is, of course, true.
You cannot, of course, expect uniform convergence (of the series of derivatives) on all of of $\mathbb{R}$