Proving a function of bounded variation is Riemann Integrable

In summary, to show that a function f is Riemann integrable on [a,b], it must be of bounded variation and its Riemann sums (S(P) and s(P)) must have a difference less than a given epsilon value. This can be achieved by setting up two partitions, one regular and one with specific criteria, and using the resulting sums to prove the desired inequality.
  • #1
ECmathstudent
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Homework Statement


If a function f is of bounded variation on [a,b], show it is Riemann integrable


Homework Equations


Have proven f to be bounded

S(P) is the suprenum of the set of Riemann integrals of a partition (Let's say J)
s(P) is the infinum of J

S(P) - s(P) < e implies f is Riemann integrable

The Attempt at a Solution



I know I'm supposed to set up two partitions, one regular one and one so that for each second piece of the partition:
f(a i) >= Mi - c
f(a i+1) =< mi -c

This will give a new sum, and I'm supposed to use this to show that S(P) - s(P) is less than epsilon. Sorry I don't know how to use Latex!
 
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  • #2
A nevermind, I've got it.
 

1. What does it mean for a function to be of bounded variation?

A function of bounded variation is one where the variation, or total amount of change, in the function over a given interval is finite. This means that the function does not have large and unpredictable fluctuations, and is generally smooth and well-behaved.

2. How is bounded variation related to Riemann integration?

A function of bounded variation is a necessary condition for Riemann integration. This means that if a function is not of bounded variation, it cannot be Riemann integrable. However, being of bounded variation does not guarantee Riemann integrability.

3. Can a function with discontinuities be of bounded variation?

Yes, a function can still be of bounded variation even if it has discontinuities. As long as the total variation over a given interval is finite, the function is considered to be of bounded variation.

4. What is the relation between bounded variation and the Riemann integral?

The Riemann integral is defined as the limit of Riemann sums, which are approximations of the area under a curve. The total variation of a function is essentially the sum of the absolute differences between the function values at different points. Therefore, a function of bounded variation has a finite Riemann integral.

5. How can one prove that a function of bounded variation is Riemann integrable?

In order to prove that a function of bounded variation is Riemann integrable, one must show that it satisfies the necessary and sufficient conditions for Riemann integrability. This includes showing that it is bounded, has a finite set of discontinuities, and that the set of discontinuities has measure zero. Additionally, one must show that the function satisfies the Cauchy criterion, meaning that the Riemann sums converge to a finite limit as the size of the subintervals approaches zero.

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