# A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of trouble with showing it's not uniformly equicontinuous.

I've tried numericals, but I can't seem to find and $\epsilon$>0 s.t. $\forall \delta >0$, $|x-y|<\delta$ and $|f_n(x) - f_m(y)|< \epsilon$. And I'm not sure how to start this proof. But I know that $\frac{n}{n+1} \cos(x^2)$ is not uniformly continuous because its derivative is unbounded.

Any help would be greatly appreciated.

• The derivative of your $f_n$ is $f_n'(x)=-\frac {2nx}{n+1}\,\sin(x^2)$, which is uniformly bounded. – Martin Argerami Nov 17 '14 at 3:26
• Perhaps I'm mistaken but I think $f_{n}:(0,\infty)\to\mathbb{R}$ defined by $f_{n}(x)=x^{\frac{1}{n}}$ works. – user71352 Dec 25 '16 at 2:30