Help with a proof on integrablitiy

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In summary, the problem is to show that an increasing function on a compact set is integrable. The definition of integrable being used is that the upper and lower integrals are equal. Since the set is compact, the function is bounded and the upper and lower integrals exist. The key is to use the fact that the function is increasing to show that the upper and lower integrals are equal. This can be done by using the properties of functions with a countable number of discontinuities and compact intervals.
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Homework Statement


Let f:[a,b] [tex]\rightarrow[/tex]R be increasing on the set [a,b] (i.e., f(x)[tex]\leq[/tex] f(y) whenever x<y. Show that f is integrable on [a,b]

Homework Equations


the definition of integrable that we are using is that [tex]\int[/tex] f=U(f)=L(f)

The Attempt at a Solution


What i tried was to start with the fact that we are on an increasing set, which is also compact. I thought since it is compact we know that we are bounded. but then i didn't know how to relate the fact that we are bounded. My thought was that since we have to use U(f) and L(f) which relate to the sup and inf, these must exist since we are on a compact set. I got here and then didn't know where to get that U(f)=L(f) So then i thought i was going in the completely wrong direction, and didn't know where else to go.
 
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  • #2
I believe that there is an easy way to get around this by using certain facts about functions with countable number of discontinuties and compact intervals.
 

1. What does it mean for a function to be integrable?

Integrability refers to a property of a function that allows it to be evaluated using integration. A function is considered integrable if it can be written in terms of a definite integral, either analytically or numerically.

2. How do I prove the integrability of a function?

To prove the integrability of a function, you must show that it meets the criteria for integrability. This includes being continuous on a closed interval and having a finite number of discontinuities. You can also use the Riemann integral or the Lebesgue integral to prove integrability.

3. Can a function be integrable on one interval but not on another?

Yes, a function can be integrable on one interval but not on another. This is because the criteria for integrability may be met on one interval but not on another. For example, a function may be continuous on one interval but have a discontinuity on another interval.

4. What is the difference between Riemann integrability and Lebesgue integrability?

Riemann integrability and Lebesgue integrability are two different methods for evaluating integrals. Riemann integrability is based on dividing the interval into smaller subintervals and approximating the function with rectangles, while Lebesgue integrability uses a more advanced approach based on the concept of measure. In general, Riemann integrability is easier to understand and apply, while Lebesgue integrability allows for a wider range of functions to be integrated.

5. Are there any special cases where a function is always integrable?

Yes, there are some special cases where a function is always integrable. For example, any continuous function on a closed interval is always integrable. Additionally, any monotonic function (either increasing or decreasing) on a closed interval is also always integrable. However, these are just a few examples and there are many other cases where a function may be integrable.

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