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The pointwise limit of a sequence of uniformly continuous functions needn't be continuous. I've been wondering about the converse:

Is a continuous function $f$ necessarily the pointwise limit of some sequence of uniformly continuous functions $(f_n)_{n\in\mathbb{N}}$?

I'm having trouble establishing or disproving this.

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If the domain of $f$ is $\Bbb R$, then the answer is yes. The proof is quite easy: let $f_n: \Bbb R \to \Bbb R$ defined by $$f_n(x)= \begin{cases} f(-n) & x<-n \\ f(x) & -n \le x \le n \\ f(n) & x>n \end{cases}$$ these are uniformly continuous and they pointwise converge to $f$.

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  • $\begingroup$ Why are the $f_n$ uniformly continuous (in $-n\leq x\leq n$)? We only know that $f$ is continuous. $\endgroup$
    – Vadim
    Jun 5, 2021 at 15:01
  • $\begingroup$ That's because on a compact set every continuous function is uniformly continuous. If you extend as a constant function outside the boundary of $[-n,n]$ it remains uniformly continuous. $\endgroup$
    – Crostul
    Jun 5, 2021 at 15:07
  • $\begingroup$ Of course, thanks for the clarification. $\endgroup$
    – Vadim
    Jun 5, 2021 at 15:10

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