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

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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.
    http://mathimatikoi.org/images/tolk_sign.png
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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.
    http://mathimatikoi.org/images/tolk_sign.png
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