1. Obtain the characterization of ultrametic preserving functions and functions whose composition with every ultrametric is a metric (we will call them quasi metric preserving function).
2. Obtain various kinds of examples: w-distances preserving functions which are/are not metric preserving, quasi metric preserving functions which are/are not metric preserving, and w-distances preserving functions which are/are not quasi metric preserving and vice versa.
3. Obtain properties of quasi metric preserving functions related to continuity, differentiability, convexity, concavity, and monotonicity. Then compare them to those of metric preserving functions.
4. If time permits, we will investigate more on the properties of other kinds of distance preserving functions in the similar way.
5. Give a new research topic to mathematics community.
