Is there a technical name for a mapping $g$ that satisfies $||g(x) - g(y)||^2 \nearrow$ when $||x - y||^2 \nearrow$?

Question

Is there a technical name for a mapping $g$ that satisfies $||g(x_n) - g(y_n)||^2 \nearrow$ when $||x_n - y_n||^2 \nearrow$ as $n$ increases?

This mapping preserves a relation about distance. Is there a name for this kind of relation as well?

Please help.