[Math] If a function $f:X\to X$ maps Cauchy sequences to Cauchy sequences then is $f$ uniformly continuous

Question

Let $X$ be an incomplete metric space. Does there exist examples of continuous function $f:X\to X$ which map Cauchy sequences to Cauchy sequences but is not uniformly continuous ?

Answer 1

First note that Cauchy-continuity implies continuity. (We may assume WLOG that $X$ is complete; else we consider the completion of $X$.) For if $x_n\to x$ in $X$, then $\{x_n\}$ is a Cauchy sequence, so given that $f$ is Cauchy-continuous, $\{f(x_n)\}$ is a Cauchy sequence and thus $f(x_n)\to f(x)$.

Now, if $X$ is totally bounded, then Cauchy-continuity implies uniform continuity. This follows from a metric space being compact iff it is complete and totally bounded, and the equivalence of continuity and uniform continuity in compact metric spaces.