Integrability of f^2: Show f is Integrable

In summary, the problem is to show that if a positive and bounded function f is integrable over [a,b], then its square f^2 is also integrable over [a,b]. The attempt at a solution involves using the fact that the upper and lower sums of f^2 are arbitrarily close over a partition P, and finding an upper bound for the difference between the upper and lower sums of f. It is suggested to consider cases where f(x)>=1 and f(x)<1, and to take the limit as m approaches zero. This is possible because the interval of integration is finite.
  • #1
Markjdb
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Homework Statement



Let f be positive and bounded over [a,b]. If f^2 is integrable over [a,b], then show that f is as well.

The Attempt at a Solution



I'm just trying to use the fact that the upper and lower sums of f^2 over a partition P are arbitrarily close, and then somehow find an upper bound for the difference of the upper and lower sums of f based on that. I've tried separating it into cases, where the max and min of f on an interval are >=1, <1, etc. but it hasn't really led anywhere. If anyone could give me some sort of hint, i'd really appreciate it =)
 
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  • #2
It's fairly easy if you know f(x)>=m for m>0 isn't it? Since f(x1)^2-f(x2)^2=(f(x1)-f(x2))*(f(x1)+f(x2)) and you know f(x1)+f(x2)>=2m. So f(x1)^2-f(x2)^2>=(f(x1)-f(x2))*2m. Now let m approach zero. Why can you ignore the part of the sums coming from f(x)<m? Remember the interval of integration is [a,b]. It's finite.
 
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1. What does it mean for a function to be integrable?

Integrability refers to the ability to calculate the definite integral of a function within a given interval. In other words, it means that the area under the curve of a function can be determined using a set of mathematical techniques.

2. How is integrability of f^2 different from integrability of f?

The main difference is that integrability of f^2 refers to the ability to calculate the definite integral of the squared function, while integrability of f refers to the ability to calculate the definite integral of the original function. In some cases, the integrability of f^2 may be easier to determine than the integrability of f.

3. What are the requirements for a function to be integrable?

In order for a function to be integrable, it must be continuous within the given interval and have a finite number of discontinuities. Additionally, the function must not have any infinite or undefined values within the interval.

4. What techniques can be used to show that f^2 is integrable?

One technique is to use the definition of Riemann integrability, which states that a function is integrable if its upper and lower Riemann sums approach the same value as the partition of the interval becomes finer. Another technique is to use the Fundamental Theorem of Calculus, which states that if a function is continuous and has a finite number of discontinuities within an interval, then it is integrable on that interval.

5. Can a function be integrable but not continuous?

No, a function must be continuous within the given interval in order to be integrable. If a function has a discontinuity within the interval, it may still be integrable if the discontinuity is removable or if the function is piecewise continuous.

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