A First Course in Real Analysis pp 300-321 | Cite as

# Functions of bounded variation and the Riemann-Stieltjes integral

## Abstract

In earlier chapters we developed the basic properties of functions on ℝ_{1} 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 ℝ_{1}to ℝ_{1}, 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 ℝ_{1}, 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.

## Keywords

Monotone Function Bounded Variation Prove Theorem Convergent Subsequence Riemann Integral## Preview

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