# Family of uniformly continuous functions, pointwise equicontinuous but is not uniformy equicontinuous.

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?

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 $$f_n(n) - f_n(n - 1/n) = 2 - 1/n^2 \ge \epsilon$$.