2
$\begingroup$

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.

$\endgroup$
  • 7
    $\begingroup$ What if the sequence is for all $n$, $a_n = f$ where $f$ discontinuous? $\endgroup$ – Don Thousand Nov 26 '18 at 16:51
  • $\begingroup$ 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? $\endgroup$ – Cosmic Nov 26 '18 at 16:58
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Cosmic Nov 26 '18 at 17:22
  • 1
    $\begingroup$ In the answer given, $x^n$ is not discontinuous, so I am still confused here. $\endgroup$ – Cosmic Nov 26 '18 at 17:25
  • 1
    $\begingroup$ 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 $n$. $\endgroup$ – José Carlos Santos Nov 26 '18 at 17:28
  • $\begingroup$ 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. $\endgroup$ – Cosmic Nov 26 '18 at 17:36
  • 1
    $\begingroup$ Yes, you are right. $\endgroup$ – José Carlos Santos Nov 26 '18 at 17:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.