If the limit function is differentiable, must its derivative equal the term by term derivative? A well-known theorem states that if $\frac{d}{dx} a_n(x)$ is uniformly convergent then the limit $a_n(x)\rightarrow a(x)$ can be differentiated term by term, i.e. $\frac{d}{dx}a(x) = \lim_{n\rightarrow \infty}\frac{d}{dx}a_n(x)$.
Now suppose $f(x)=\lim_{n\rightarrow \infty} f_n(x)$ converges on $\mathbb{R}$ and is differentiable. Suppose $ f_n'(x)$ also converges pointwise everywhere. Without any uniform convergence assumptions, must we have $f'= \lim_{n\to\infty} f_n'(x)$ a.e.?
I've seen an example here where they differ at one point.
I think the answer is yes since pointwise convergence of continuous functions implies uniform convergence on a set of measure as large as we want, so they at least must agree on a large set. I tried to modify the proof given in Rudin but its not clear where we can swap "uniform convergence" for "existence of $g$ derivative"
 A: Thanks to Dave Renfro (whose posts on StackExchange are legend).  In the link
he points us to this paper:
Darji, Udayan B.   Limits of differentiable functions.  Proc. Amer.
Math. Soc. 124 (1996), no. 1, 129-134.
Abstract. Suppose that $\{f_n\}$ is a sequence of differentiable functions defined on $[0,1]$ which converges uniformly to some differentiable function $f$,   and  $f_n'$  converges pointwise to some function $g$.  Let $M= \{x: f'(x)\not = g(x)\}$.  In this paper we characterize such sets $M$  under various hypotheses.  It follows from one of our characterizations that $M$  can be the entire interval $ [0,1]$.
I cannot resist quoting Dave's account of Darji's research (by the way we all call Darji "Darji").

"Interestingly, almost no research into refinements of this result
exist, at least for real-valued functions of one real variable.
Sometime around 1989, give or take a year, Udayan B. Darji asked his
Ph.D. advisor Jack B. Brown (Auburn University) if he knew of any
results in the literature that examined how different g could be from
F'. Brown didn't know of any and passed the question along to Andrew
M. Bruckner. Bruckner, perhaps after asking others (Solomon Marcus,
Jan S. Lipinski, etc.), reported back that he wasn't aware of anyone
having done anything with this topic.
I don't know how widely it was known in 1989 that this obvious
question had apparently never been addressed before, at least in
print, but by 1991 many specialists in real analysis were aware of
Darji's work and found it quite surprising that apparently no one had
previously done any work on this problem. Incidentally, these remarks
are based on my recollection of what I heard first-hand from all the
participants in this story. ... Darji's work on this topic formed a
little more than half of his 1991 Ph.D. Dissertation, and a condensed
version of this part of his Dissertation was published in 1996."

