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Thinking about the (probably) well-known fallacy about approaching a unit square diagonal with staircase functions and thus concluding the diagonal length be $2$ instead of $\sqrt 2$ led me to an interesting question:

Given a sequence $(f_k)_{k\in\mathbb N}$ of differentiable functions converging towards a differentiable limit function $f$, when does the limit of derivatives coincide with the derivative of the limit function, that is, when do we have $f'(x)=\lim_{k\to\infty}f_k'(x)$ for all $x$ in the function's domain? And what about second or $n$-th derivatives, supposing all the functions $f_k$ as well as $f$ are twice or $n$ times differentiable?

No need to tell me staircase functions aren't differentiable - this is supposed to be a more general question about necessary and sufficient conditions for the limit of $n$-th derivatives to coincide with the $n$-th derivative of the limit.

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You can have weaker hypotheses. Namely:

Suppose $(f_n)$ is a sequence of real or complex-valued differentiable functions on an interval $I\subset\mathbf R$. If

  • The sequence $(f'_n)$ converges uniformly on every bounded closed interval contained in $I$ to a function $g$,
  • $\bigl(f_n(x_0)\bigr)$ converges for at least one point $x_0\in I$,

then the sequence $(f_n)$ converges uniformly on every bounded closed interval contained in $I$ to a function $f$, differentiable on $I$, and $f'=g$.

The same result is true if ‘differentiable’ is replaced by $C^1$.

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Suppose all functions are defined on an open set.

If $f_n \to f$ pointwise and $f_n' \to g$ uniformly, then $f$ is differentiable and $f' = g$.

For higher derivatives you should have $f_n^{(m)}$ converging uniformly.

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