## 20.3 Always Integrable Functions

There are two circumstances in which we know that f will be Riemann integrable over the interval between a and b.

If f is continuous everywhere in the interval including its endpoints which are finite, then f will be integrable.

A function is continuous at x if its values sufficiently near x are as close as you choose to one another and to its value at x.

Continuity of f throughout [a, b] implies that the variation in estimates in any strip can be made as small a multiple of the width of the strip as you choose by you making the width small enough.

Thus you can make the total possible variation in Riemann sums as small a multiple of b - a as you like, by requiring that the maximum strip width is sufficiently small.

We can (and soon will) also prove that a bounded increasing or decreasing function on a finite interval is integrable even if it is not continuous.

We can define the total variation of a function to be the total of its increase over intervals in which it is increasing, plus total of its decrease over intervals in which it is decreasing.

We will deduce that a function having bounded (total) variation between a and b will be Riemann integrable in that interval.