Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous

Question

Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous?

we know uniformorly continuous function maps cauchy sequence to cauchy sequence.But my book doesn't say anything about converse of i think its false. but didn't find a counter example yet . please help!

Answer 1

Since, in $\mathbb R$, a sequence is a Cauchy sequence if and only if it converges, what you are asking for is a non-uniformly continuous map from $\mathbb R$ into itself which maps convergent sequences into convergent sequences. Any continuous but non-uniformly continuous map will do that. Take any polynomial function whose degree is at least $2$, for instance.

Answer 2

Every continuous map on $\mathbb{R}$ sends a Cauchy sequence to a Cauchy sequence (because Cauchy sequences convergence).

So you can pick any continuous but non uniformly continuous map on $\mathbb{R}$ as a counter-example. Say $f(x)=x^2$.