Math Is Fun Forum

GOKILL

Member

Registered: 2010-03-19

Posts: 26

still riemann integral

#2
Given

and

Let

Prove that

i think when the function f is monotonic increasing,that can be done, but otherwise?

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zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,432

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Re: still riemann integral

3 years late, but might be useful to people looking up the same thing.

First, note that your second partition is a refinement of the first with one extra point -- namely, y. I prove the general case for any two partitions P and P* where P* is a refinement of P, so that your result follows as an immediate corollary.

Let P be a partition of [a,b], with U(f,P) and L(f,P) being the upper and lower Darboux sums of f with respect to the partition P. Write the original partition P as

and the partition P* as

where x* is the extra point. Then:

But note that

and

substituting, we see that the RHS of the inequality is greater than or equal to 0, thus

.

The inequality

is done similarly. Thus, by repeated application of this proof, this holds true for any refinement of P. Using this result, let

be a common refinement of P and Q. Then:

and

but, obviously

hence

QED.

Last edited by zetafunc (2014-05-25 19:12:41)