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Theorem: If $f_{n}$ is a sequence of differentiable functions on $[a,b]$ such that $\lim _{n\to \infty }f_{n}(x_{0})$ exists (and is finite) for some $x_{0}\in [a,b]$ and the sequence $f'_{n}$ converges uniformly on $[a,b]$, then $f_{n}$ converges uniformly to a function $f$ on $[a,b]$, and $f'(x)=\lim _{n\to \infty }f'_{n}(x)$ for $x\in [a,b]$.

I have known how to prove the theorem when $fn$ has a continuous derivative because I can use fundamental theorem of calculus to construct a function that is $$f(x_0)+\int_{x_0}^xg(t)dt$$ where $g(t)$ is the limit that $f'_n$ is convergent to.

But I have read this in Wikipedia that the derivative needn't be continuous, which means fundamental theorem of calculus fails to work.

So my question is how to prove the theorem without the contition that the derivative is continuous.

nyan
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    You can use mathjax viewer for your test with LaTeX. – user1851281 Apr 30 '22 at 08:24
  • @user281 Thank you so much. And I have edit my question so that it is initiated formally. – nyan Apr 30 '22 at 08:27
  • This one gets asked a lot. Study the answers here: https://math.stackexchange.com/questions/214218/uniform-convergence-of-derivatives-tao-14-2-7?rq=1 – B. S. Thomson Apr 30 '22 at 16:15
  • The natural follow-up question is what can happen if the convergence of the sequence of derivatives ${f_n'}$ is not uniform, just pointwise. For an answer go here: https://math.stackexchange.com/questions/4331951/if-the-limit-function-is-differentiable-must-its-derivative-equal-the-term-by-t/4332004#4332004 – B. S. Thomson Apr 30 '22 at 16:20
  • @B. S. Thomson Thank you! That's much useful for me. – nyan May 01 '22 at 01:57

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Try to apply the mean value theorem in $[x_{0},x]\subset[a.b]$, with x variable, to $f_{m}-f_{n}$ for some $n.m\in \mathbb{N}$. Then apply the norm both sides and your hypothesis to prove that $f$ converges uniformly. The next step is similar, apply MVT in $[c,x_{0}]$ to the same difference.

Guillermo García Sáez
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