2
$\begingroup$

Let $(g_n)$ be a sequence of functions on $[a,b]$. If $(g_n)$ converges pointwise and uniformly continuous on $[a,b]$, does $(g_n)$ also converge uniformly on $[a,b]$?

$\endgroup$
3
$\begingroup$

Take $g_n(x)=x^n;x\in [0,1]$

Every continuous function on a compact set is uniformly continuous

But $\lim_{n\to \infty} g_n(x)=g(x)=0 ;0\le x<1 \text{and} 1,x=1$

which is not even continuous

| cite | improve this answer | |
$\endgroup$

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.