I want to find a family of uniformly continuous functions $\{f_{n}\}$ such that $\{f_{n}\}$ is pointwise equicontinuous but is not uniformy equicontinuous.
I'm having trouble finding an explicit example. I saw this answer: https://math.stackexchange.com/a/2594576/444015, but I dont understand why the condition (ii) implies $F$ not uniformly equicontinuous.
Can someone help me?
Best Answer
Try $$ f_n(x) = \cases{x^2 & if $|x| \le n$\cr n^2 & otherwise\cr} $$
Each $f_n$ is uniformly continuous: $|x - y| < \epsilon/(2 n)$ implies $|f_n(x) - f_n(y)| < \epsilon$.
The sequence is equicontinuous at every $x$: $|x - y| < \min(1, \epsilon/(2+2|x|))$ implies $|f_n(x)-f_n(y)| < \epsilon$.
The sequence is not uniformly equicontinuous: take $\epsilon = 1$, and note that $f_n(n) - f_n(n - 1/n) = 2 - 1/n^2 \ge \epsilon$.