Riemann's Integrability Condition

In summary, the conversation is about a proof from a link provided by one person to another. The proof can be found on pages 8-9 and the person is struggling to understand an assertion made by the author on the top of page 9. The assertion states that $$0 \le U(f) - L(f) \le U(f,P) - L(f;P) < \epsilon$$ and the person is questioning the truth of this statement and looking for help to understand it. They mention a previous proposition in the proof that may have been overlooked.
  • #1
Bashyboy
1,421
5

Homework Statement


Here is a link to the proof I am reading: https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf

Homework Equations

The Attempt at a Solution



The proof to which I am referring can be found on pages 8-9. At the top of page 9, the author makes an assertion which I endeavored to account for, but have been unsuccessful. Here is the assertion:

$$0 \le U(f) - L(f) \le U(f,P - L(f;P) < \epsilon$$

Specifically, I am referring to $$U(f) - L(f) \ge 0$$. Is this really true; how do they know it will always be zero or positive? I have tried to justify it, but have failed. Could someone possibly help me?
 
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  • #2
Am I misunderstanding, or did you happen to oversee proposition 1.13 and its proof, just above section 1.4?
 

Related to Riemann's Integrability Condition

1. What is Riemann's Integrability Condition?

Riemann's Integrability Condition is a mathematical concept that determines whether a function is integrable or not. It is based on the idea that a function is integrable if and only if the upper and lower Riemann sums converge to the same value as the partition of the interval becomes infinitely small.

2. How is Riemann's Integrability Condition used?

Riemann's Integrability Condition is used to determine whether a function can be integrated using the Riemann integral. It is also used to determine the convergence of improper integrals.

3. What are the necessary conditions for a function to be Riemann integrable?

The necessary conditions for a function to be Riemann integrable are that the function must be bounded on the interval of integration and it must have a finite number of discontinuities within that interval.

4. What is the difference between Riemann's Integrability Condition and the definition of the Riemann integral?

While Riemann's Integrability Condition is a necessary condition for a function to be integrable, the definition of the Riemann integral also requires the function to be continuous on the interval of integration. In other words, a function can satisfy Riemann's Integrability Condition but still not be Riemann integrable if it has a discontinuity on the interval.

5. Can Riemann's Integrability Condition be applied to all functions?

No, Riemann's Integrability Condition can only be applied to bounded functions on a closed interval. Functions that are unbounded or have an infinite number of discontinuities within the interval cannot be evaluated using Riemann's Integrability Condition.

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