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?