Traditionally, equations with discontinuities in space variables follow the ideology of the sliding mode'. This book contains the first account of the theory which allows the consideration of exact solutions for such equations. The difference between the two approaches is illustrated by scalar equations of the type y =f(y) and by equations arising under the synthesis of optimal control. A detailed study of topological effects related to limit passages in ordinary differential equations widens the theory for the case of equations with continuous right-hand sides, and makes it possible to work...