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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 interval 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
change over the intervals in which it is increasing, plus the absolute value of
its total change over intervals in which it is decreasing. Then we can deduce
that a function having bounded (total) variation between a and b will be
Riemann integrable in that interval.
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