Which bounded functions are integrable?

This is a difficult question to answer. One simple partial answer is that monotonic functions are integrable.

A function is monotonic on an interval if it is either increasing or decreasing on the interval. If f is a monotonic function on [a,b] then it is bounded and integrable. (And the result can be easily generalized to bounded piecewise monotonic functions [see Ross, p. 196]).

"Fortunately, most of the functions that occur in practice are monotonic or sums of monotonic functions, so the result of this miniature theory of integration is quite comprehensive." (Apostol, p. 76)

Before considering the theoretical aspect of the problem we can think a little about the approximation of the value of the definite integral of a monotonic function.

When approximating an integral using rectangles we make an error. In some cases, for example, in monotonic functions, we can limit the magnitude of the error. The error is equal to the sum of the areas of the blue "curved triangles".

This error is less than the area of a rectangle.

If we consider a simple case, when the bases of the rectangles are equal, then we can calculate an error bound:

Using refined partitions we obtain as small bounds of the error as we want. Because when we refine a partition the base of the rectangle is smaller.

Now we are going to prove that a monotonic positive increasing function on [a,b] is integrable.

We start with a partition of the interval:

If we divide the interval into n equal parts then the width of each rectangle is:

We can calculate an upper bound of the integral:

We consider the decreasing sequence

With the same partition, this is a lower bound of the integral:

We consider the increasing sequence

If this two sequences converge toward one and the same limit, we can call this limit the definite integral, and write

But the question always is whether they do converge.

When we consider an increasing monotonic function we have an increasing sequence (S_n) and a decreasing sequence (T_n). Then the question is only whether we can verify

We can calculate T_n - S_n

Then the limit

We conclude that if f is monotonic in the interval [a,b], then the definite integral exists. [Toeplitz, p.64]

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