## Example of sequence of functions [entire topic moved to new forum]

Grigorios Kostakos on Sunday, October 25 2015, 03:07 AM
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Give an example of a sequence of discontinuous functions which converges uniformly to a continuous function.

Grigorios Kostakos

• Replied by Apostolos J. Kos on Sunday, October 25 2015, 10:03 AM · Hide · #1
$$f_n(x)= \left\{\begin{matrix} 0 & , &x\neq 0 \\ 1/n &, & x=0 \end{matrix}\right., \;\;\;\;\;\;\; g_n(x)= \left\{\begin{matrix} 1/n & , & x \in \mathbb{Q}\\ 0&, &{\rm elsewhere} \end{matrix}\right.$$

Verification immediate.
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Replied by Apostolos J. Kos on Monday, November 02 2015, 11:39 AM · Hide · #2
I don't want to open a new thread, so I post this interesting question here.

Give an example of a sequence of infinitely differentiable functions that converge uniformly to zero but whose the sequences of derivatives diverge everywhere.