Functions of bounded variation and the Riemann-Stieltjes integral

Abstract

In earlier chapters we developed the basic properties of functions on ℝ₁ and ℝ _n Of particular interest are conditions which determine when functions are continuous, differentiable of any order, analytic, and integrable. There are many problems, especially in the applications to physical sciences, in which we require more precise information than we have obtained so far about the behavior of functions. In the simplest case of functions from ℝ₁to ℝ₁, it is useful, for example, to be able to measure how rapidly a function oscillates. However, the oscillatory character of a function is not easily determined from its continuity or differentiability properties. For this reason, we introduce the notion of the variation of a function, defined below. This quantity turns out to be useful for problems in physics, engineering, probability theory, Fourier series, and so forth. In this section we establish the principal theorems concerning the variation of a function on ℝ₁, and in Section 12.2 we show that this concept can be used to define an important extension of the Riemann integral, one which enlarges substantially the class of functions which may be integrated.