# Uniform convergence and equicontinuity

Given a sequence of functions which is not uniformly convergent, can we deduce, that none of its subsequences is uniformly continous and therefore, by Arzela-Ascoli say that the family of function is not equicontinous?

I think it is true in the case that the limit function is not continous (because all the subsequence must converge pointwise to that function, and then the convergence cannot be uniform). But what when the limit is continous?

• You seem to be asking if there is a sequence of functions that isn't equicontinuous converging to a continuous limit? The answer is obviously yes. You can approximate the straight line (in a pointwise sense) $f(x) = x$ very easily using functions that aren't even continuous themselves (let alone being equicontinuous as sequence). – Frank Apr 5 '15 at 16:19
• Suppose I have the sequence of functions $\{f_{n}\}$ where $f_{n}(x)=x^n, x\in [0,1].$. It is easy to see that its limit function is not continous. Therefore, we cannot have uniform convergence. Can I say that all the subsequences of $\{f_{n}\}$ cannot be uniformly convergent ? – Bill Apr 5 '15 at 16:31
• The answer to your first question is obviously no. Define $f_n(x)=(-1)^n$ for all $x$. This is not (uniformly) convergent, but it is equicontinuous. – Vincent Boelens Apr 5 '15 at 16:31