An inverse question of uniformly convergence

Question

{Edit: since I made some mistake on the pointwise limit and the uniformly continuous.}

A classical results in elementary analysis state that if a sequence of continuous function $f_n(x)$ on $[0,1]$ is uniformly convergence to $f$, then $f$ is continuous on $[0,1]$ too.

I am wondering that if we know that $f$ is continuous on $[0,1]$, and it is a pointwise limit of a sequence of continuous function $f_n$ on $[0,1]$, can we conclude that $f_n$ convergence to $f$ uniformly on $[0,1]$?

I have noted that if $f$ is not continuous, then there has a counter example, $f_n=x^n$, $f(1)=1$ and $f=0$ otherwise. But how about add the continuous to $f$?

Your first paragraph is wrong. The pointwise limit of a sequence of continuous functions need not be continuous. — Daniel Fischer, Oct 27 '13 at 13:28
For a less extreme example consider the functions $f_n$, where the graph of $f_n$ consists of the straight line segments connecting the points $(0,0)$, $(1/n,1)$, $(2/n,0)$, and $(1,0)$. — David Mitra, Oct 27 '13 at 13:32
@DanielFischer Thanks, I made some mistake on uniformly continuous. — van abel, Oct 27 '13 at 14:17

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

No. For example, consider the functions $f_n$ on $[0,1]$, where the graph of $f_n$ consists of the straight line segments connecting the points $(0,0)$, $(1/(n+1),1)$, $(1/n,0)$, and $(1,0)$. The sequence of continuous functions $(f_n)$ converges pointwise to the zero function on $[0,1]$; but, the convergence is not uniform.

More interesting, is that a pointwise convergent sequence of continuous functions to a continuous limit need not converge uniformly on any open subset of $[0,1]$. See this post for counterexamples.

An inverse question of uniformly convergence

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