The very first example of pointwise convergence of continuous functions one meets is $f_n(x)=x^n$ with $x\in [0,1]$.

Here $f_n$'s are continuous on $[0,1]$, but their pointwise limit is not continuous at $1$.

Q. Is there an example of sequence $f_n$ of continuous functions from $[0,1]$ to $\mathbb{R}$ such that their pointwise limit is nowhere continuous?

  • $\begingroup$ I think the Dirichlet function would help: en.wikipedia.org/wiki/Nowhere_continuous_function $\endgroup$
    – ntt
    Apr 10 '17 at 5:23
  • $\begingroup$ No, the limit function will be continuous on a dense $G_\delta$ subset of $[0,1]$ (which must be uncountable btw). This is a consequence of Baire's theorem. $\endgroup$
    – zhw.
    Apr 10 '17 at 5:29

$$f_n(x)=\begin{cases} 1+\dfrac{x}{n}, \: x\in [0,1]\cap\mathbb{Q}\\0,\: \text{otherwise}\end{cases}$$

  • $\begingroup$ Is $f_n$ continuous on whole $[0,1]$? $\endgroup$
    – Beginner
    Apr 10 '17 at 5:30
  • 1
    $\begingroup$ I didn't check that. I will get back to you. I will update the answer soon. $\endgroup$
    – creative
    Apr 10 '17 at 5:36
  • $\begingroup$ Obviously not by density of the rationals and irrationals $\endgroup$
    – James_T
    Jul 13 '20 at 17:31

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