1 - Distribution function

Abstract

This chapter describes the different aspects of distribution function. It presents an assumption where f is an increasing function defined on the real line (−∞, +∞). For any two real numbers x1 and x2, x1< x2→ f(x1) ≤ f(x2), the only possible kind of discontinuity of an increasing function is a jump. It is observed that points of jump may have a finite point of accumulation and that such a point of accumulation need not be a point of jump itself. Thus, the set of points of jump is not necessarily a closed set. It is shown that the set of points of jump of an increasing function may be everywhere dense and the set of rational numbers in the example may be replaced by an arbitrary countable set without any change of the argument. It is found that the condition of countability is indispensable.