A function $f$ is metric preserving if for every metric space $(M,\rho)$ we have that $f \circ \rho$ is still a metric on $M$. In this article we look at the behavior of such functions with respect to continuity and differentiability. We include several pathological examples and some open questions.
"Continuity and Differentiability Aspects of Metric Preserving Functions." Real Anal. Exchange 25 (2) 849 - 868, 1999/2000.