Functions preserving metrics and Gromov-Hausdorf space
The paper is focused on the study of deformations of metric spaces induced by metric preserving functions. We show that continuous metric preserving functions correctly define maps of the Gromov--Hausdorff space to themselves, and these maps have several interesting properties, in particular, they are continuous and they are Lipschitzian if and only if the corresponding metric preserving functions are Lipschitzian. We also study one-parameter deformations of arbitrary metrics defined by metric preserving functions and provide a criterion for the continuity of lengths of curves under such deformations.