If the pointwise limit is uniformly continuous, the functions in the sequence need not be so

Question

If $f_{n}$ is a sequence of uniformly continuous functions and $f_n \to f$, then $f$ is a continuous function.

Why is the converse of this statement not necessary true? Is there a simple example?

Thanks.

Answer 1

Consider the functions $f_n=\chi _{[-n,n]}$. then the limit is the function $f=1$ which is continuous, but clearly $f_n$ does not converge uniformly.

Answer 2

The Cantor Function is a good counterexample, in its alternative definition here:

Cantor Function in Wikipedia

Answer 3

For positive integer $n$, let $f_n(x)=0$ for $x<1/n$ and for $x>3/n.$ For $1/n\leq x \leq 2/n$ let $f_n(x)= n^2 (x-1/n).$ For $2/n<x\leq 3/n$ let $f_n(x)=n^2 (-x+3/n).$ We have $\lim_{n\to \infty}f_n(x)=0$ for every $x$, but $f_n(2/n)=n$ for every $n,$ so convergence of $(f_n)_{n\in N}$ to $0$ is not uniform.