Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

Question

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either:

• Proofs i.e. pointwise equicontinuity is uniform equicontinuity provided the domain is compact

• No distinction made whatsoever between pointwise and uniform equicontinuity

• Only used to prove Arzela Ascoli and that's the end of the conversation on equicontinuity

Can someone please provide a concrete example of a sequence of function that is pointwise equicontinuous but not uniform equicontinuous?

Or some examples of what pointwise equicontinuous sequences and some examples of uniform equicontinuous sequences? I hope I am not asking too much.

The only example I can think of is the trivial example: $f_n(x) = n$, but sequence is both pointwise and uniform equicontinuous so It doesn't really shine a light on the difference between the two concepts

Answer 1

Take your favourite example of a function $f:\mathbb R\to\mathbb R$ which is continuous but not uniformly continuous, e.g. $f(x)=x^2$, and then define $f_n:=f$ for all $n\in\mathbb N$. This gives you a sequence $(f_n)$ which is pointwise equicontinuous (by the very definition of pointwise equicontinuity), but not uniformly equicontinuous (also by the very definition of uniform equicontinuity).