On the evolution of set-valued functions: An example

Within set-valued analysis there exist several notions related to the infinitesimal behavior of set-valued functions. One more such notion, needed for a class of spacial statistical problems, was proposed in Khmaladze (2007). Recently we found another suitable characteristic of this evolution, which has a form very similar to what is known as a failure rate in reliability theory and survival analysis, and which was used to describe evolution in a lily pond-type process. In this note we show through an example that there are infinitely many set-valued functions with the same failure rate-type characteristic, but all with different set-valued derivatives of Khmaladze (2007).