Prove that this function is Riemann integrable

In summary, the author is having trouble proving that a function is Riemann integrable on a given interval. However, by showing that the terms of the sequence converge to zero, they may be able to prove continuity.
  • #1
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Let f:[a,b]->R be bounded. Further, let it be continuous on [a,b] except at points a1, a2, ...,an,... such that a1>a2>a3>...>an>...> a where an converges to a. Prove that f is Riemann integrable on [a,b].

It suffices to prove that f is integrable on [a,a1) (I've worked out that part). And that's what I'm having trouble with.
 
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  • #2
Are you dealing strictly with Riemann integration, or is this proof done in the context of Lebesgue integration?

One might start out by noting that:

[tex]\int_{a}^{b} f(x)dx = \int_{a_1}^{b} f(x)dx + \lim_{n\rightarrow\infty}\sum_{k=2}^{n} \int_{a_{k-1}}^{a_{k}} f(x)dx [/tex]
 
  • #3
Riemann. And I had already figured out that part. What I'm having trouble with is that I must show the terms [tex]\int_{a_n}^{a_{n+1}}f \rightarrow 0[/tex] as [tex]n\rightarrow \infty[/tex]. Since the series is bounded, showing that the terms converge to 0 will imply the convergence of the series.
 
  • #4
What is the definition of convergence of a sequence, say a_n ?
 
  • #5
If I got by antiderivatives, that is F(a_n+1)-F(a_n), there's no way to garantee the continuity of F (where F' = f). However since F is bounded by M, the integral is bounded below and above by +/-M(a_n+1 - a_n) which squeezes the integral to zero by convergence of a_n.
 
  • #6
So you have an infinite sequence of points an[/b] that converge to a such that f is discontinuous at those points? You need more than that: you will also need that there is only a finite step discontinuity at those points.
 
  • #7
I found a flaw in my argument. If a series is bounded, and the terms converge to zero, it DOES NOT imply the convergence of the series. I'll have to try another approach.
 

1. What is the definition of Riemann integrability?

Riemann integrability is a mathematical concept that describes the ability of a function to be integrated over a specific interval. A function is considered Riemann integrable if it can be divided into smaller subintervals, and the limit of the upper and lower sums of these subintervals approaches the same value as the subintervals shrink towards zero.

2. How can I determine if a function is Riemann integrable?

To determine if a function is Riemann integrable, you can use the Riemann integral test. This involves calculating the upper and lower sums of the function over a given interval and checking if the limit of these sums exists and is equal. If it does, then the function is Riemann integrable over that interval.

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

If a function is Riemann integrable, it means that the area under the curve of the function can be calculated using the Riemann integral formula. This allows for the calculation of definite integrals, which have many practical applications in mathematics and science.

4. Can all functions be proven to be Riemann integrable?

No, not all functions are Riemann integrable. A function must meet specific criteria, such as being bounded and having a finite number of discontinuities, to be considered Riemann integrable. Some functions, such as those with infinite discontinuities or unbounded functions, are not Riemann integrable.

5. Are there any other types of integrability besides Riemann integrability?

Yes, there are other types of integrability, such as Lebesgue integrability and improper integrability. These types of integrability have different criteria and methods for determining if a function is integrable. Riemann integrability is one of the most commonly used and studied types of integrability.

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