We know that if a function $f: A \mapsto \mathbb{R}$, $A \subseteq \mathbb{R}$, is uniformly continuous on $A$ then, if $(x_n)$ is a Cauchy sequence in $A$, then $(f(x_n))$ is also a Cauchy sequence.
I would like an example of continuous function $g: A \mapsto \mathbb{R}$ such that for a Cauchy sequence $(x_n)$ in $A$, it is not true that $f(x_n)$ is a Cauchy sequence.
Thanks for your help.