Discontinuous Functions as Limits of Compactly Supported Formulas

A bounded real valued function with domain R and one point of discontinuity can be discontinuous in six ways. In beginning textbooks such functions are usually defined piecewise with each piece being given by a formula. Here we give six examples, each having a different type of discontinuity at its unique point of discontinuity. Each example type is represented as a pointwise limit of quite simple continuous functions. Each approximating function can be given by an elementary formula and also can be chosen to be of compact support.