Here's a standard textbook question:
If $f_n(x)\to f(x)$ pointwise at any $x\in\mathbb{R}$ and $f_n,f$ are all differentiable, does $\lim f_n'(x)=f'(x)$ for all $x$ ?
The answer is 'no', even the limit may not exist at some points, and the standard counterexample is $f_n(x)=\frac{\sin nx}{n}$.
Now my question (also inspired by Term-by-term differentiation of a sequence of functions without uniform convergence of derivatives):
Assume that
$f_n(x)\to f(x)$ pointwise at any $x\in\mathbb{R}$
$f_n,f$ are all differentiable at any $x\in\mathbb{R}$
$f_n'(x)\to g(x)$ pointwise at any $x\in\mathbb{R}$ (this eliminates the standard examples like above)
Does it imply $g(x)=f'(x)$ for all $x$ ?
Probably the answer is still 'no' (and 'yes' for uniform convergence), but I couldn't find a counterexample here.
