Home | 18.01 | Chapter 12 | Section 12.3

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The "variation" of a function is the sum of its rises where it is increasing and its declines where it decreases.

In the previous comment, we proved:

Theorem 1: If f has bounded variation in an interval, its integral over that interval exists and is well defined.

A slightly stronger statement, also true, is:

Theorem 2: If f is continuous over a closed interval [a,b], it is integrable over it.