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- I am comparing two theorems in two different texts on uniform convergence and differentiability; the first one is from some lecture notes, the second from Rudin's PMA. Apart from working on open versus closed intervals, what is the difference between these two theorems?

The first theorem is from here (page 9 in the pdf):

The second theorem is from baby Rudin:

Obviously the open/closed intervals is one difference, but if we were to replace the open intervals with closed intervals, is the first theorem a special case of the second one? Some have said they are even roughly equivalent, which I don't see. The way I see it is that the first theorem assumes ##f_n(x)## converges for all ##x## in ##(a,b)## (or ##[a,b]## if we were to replace the open intervals with closed intervals), whereas the second theorem only assumes ##f_n(x)## converges for a single point. I'll be honest and say I have read neither proofs so far. What I have read is a proof under additional assumptions (continuity of ##f_n##), but that's another theorem. Any comments are appreciated.

Theorem 9.18.Suppose that ##(f_n)## is a sequence of differentiable functions ##f_n:(a,b)\to\mathbb R## such that ##f_n\to f## pointwise and ##f_n'\to g## uniformly for some ##f,g:(a,b)\to\mathbb R##. Then ##f## is differentiable on ##(a,b)## and ##f'=g##.

The second theorem is from baby Rudin:

7.17 TheoremSuppose ##\{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 $$f'(x)=\lim_{n\to\infty}f_n'(x)\quad (a\leq x\leq b).$$

Obviously the open/closed intervals is one difference, but if we were to replace the open intervals with closed intervals, is the first theorem a special case of the second one? Some have said they are even roughly equivalent, which I don't see. The way I see it is that the first theorem assumes ##f_n(x)## converges for all ##x## in ##(a,b)## (or ##[a,b]## if we were to replace the open intervals with closed intervals), whereas the second theorem only assumes ##f_n(x)## converges for a single point. I'll be honest and say I have read neither proofs so far. What I have read is a proof under additional assumptions (continuity of ##f_n##), but that's another theorem. Any comments are appreciated.