Introduction

Nowadays, many mathematicians agree that discontinuity as well as continuity should be considered when one seeks to describe the real world more adequately. The idea that, besides continuity, discontinuity is a property of motion is as old as the idea of motion itself. This understanding was strong in ancient Greece. For example, it was expressed in paradoxes of Zeno. Invention of calculus by Newton and Leibniz in its last form, and the development of the analysis adjunct to celestial mechanics, which was stimulating for the founders of the theory of dynamical systems, took us away from the concept of discontinuity. The domination of continuous dynamics, and also smooth dynamics, has been apparent for a long time. However, the application of differential equations in mechanics, electronics, biology, neural networks, medicine, and social sciences often necessitates the introduction of discontinuity, as either abrupt interruptions of an elsewhere continuous process (impulsive differential equations) or in the form of discrete time setting (difference equations). If difference equations may be considered as an instrument of investigation of continuous motion through, for example, Poincaré maps, impulsive differential equations seem appropriate for modeling motions where continuous changes are mixed with impact type changes in equal proportion. Recently, it is becoming clear that to discuss real world systems that (1) exist for a long period of time, or (2) are multidimensional, with a large number of dependent variables, researchers resort to differential equations with: (1) discontinuous trajectories (impulsive differential equations); (2) switching in the right-hand side (differential equations with discontinuous right-hand side); (3) some coordinates ruled by discrete equations (hybrid systems); (4) disconnected domains of existence of solutions (time scale differential equations), where these properties may be combined in a single model.