# Can a sequence of discontinuous functions converge uniformly to a discontinuous function?

I have read the theorem which says:

If continuous sequence $$(f_n(x))$$ converges uniformly to function $$f(x)$$ in some interval of real numbers, than $$f(x)$$ must be also continuous.

I was just wondering if the statement written in the question is true or not?

because if the proposed statement is not true ,then i can always say that if $$f(x)$$ comes out to be discontinuous then,$$f_m(x)$$ does not converge uniformly.

• What if the sequence is for all $n$, $a_n = f$ where $f$ discontinuous? – Don Thousand Nov 26 '18 at 16:51
• so that means that we can't say that if $f(x)$ is found to be discontinuous, then $f_m(x)$ does not converge uniformly? – Cosmic Nov 26 '18 at 16:58

The statement is true and your conclusion is correct. Take, for instance, for each natural $$n$$,$$\begin{array}{rccc}f_n\colon&[0,1]&\longrightarrow&\mathbb R\\&x&\mapsto&x^n.\end{array}$$Then, if you define$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}0&\text{ if }x<1\\1&\text{ otherwise,}\end{cases}\end{array}$$you have$$\bigl(\forall x\in[0,1]\bigr):\lim_{n\to\infty}f_n(x)=f(x).$$So, since each $$f_n$$ is continuous and $$f$$ is discontinuous, the convergence cannot possibly be uniform.

• If $f_m(x)$ converges pointwise to $f(x)$ and i find $f(x)$ to be discontinuous, then can I surely say that $f_m(x)$ does not converge uniformly to $f(x)$??That's why i asked beacause the theorem stated above does not say anything about $f_m(x)$ being discontinuous. – Cosmic Nov 26 '18 at 17:22
• In the answer given, $x^n$ is not discontinuous, so I am still confused here. – Cosmic Nov 26 '18 at 17:25
• You wrote, at the end of your question, “if the proposed statement is not true, then I can always say that if $f(x)$ comes out to be discontinuous then $f_m(x)$ does not converge uniformly”. I was addressing this conclusion. But, yes, a sequence of discontinuous functions can converge uniformly to a discontinuous function. Just take a discontinuous function $f$ and define $f_n=f$ for each natural $n$. – José Carlos Santos Nov 26 '18 at 17:28
• Thanks, But then my statement at the end of the 'I can always say that if $f(x)$ comes out to be discontinuous then,$f_m(x)$ does not converge uniformly' is NOT true , right?. for example if i take$f_m$ $=$ $f$ , where $f$ is discontinuous, then although $f$ is discontinuous, but $f_m$ still CONVERGES UNIFORMLY. – Cosmic Nov 26 '18 at 17:36
• Yes, you are right. – José Carlos Santos Nov 26 '18 at 17:38