Proving that a function is Riemann integrable

In summary, we are asked to prove that the function f is Riemann integrable. To do so, we need to show that the upper sum L(f,P) and lower sum L(f,P), for any partition P, differ by less than a given e>0. We can take advantage of the fact that the sequence a_n converges to 0 and is strictly decreasing. This means that for n>N, a_n can be less than 1/k where k is any positive number. By using this knowledge, we can produce an upper bound for U(f,P) as the norm of the partition goes to zero, allowing us to refine the inequality and prove that f is indeed Riemann integrable.
  • #1
Treadstone 71
275
0
"Let (r_n) be any list of rational numbers in [0,1]. Let (a_n) be a sequence such that 1>a_1>a_2>...>a_n>... converging to 0. Let f be defined such that

f=0 if x is irrational
=a_n if x is equal to r_x

Prove that f is Riemann integrable."

We are doing integrals from Darboux' approach, so no tagged partitions or whatnot. I have to somehow show that the upper sum L(f,P) and lower sum L(f,P), for some partition P, differ by less than a given e>0.

L(f,P) = 0 for any partition, so |U(f,P) - L(f,P)|=U(f,P). I have no idea how to proceed, because if p is a partition of n equal parts, [tex]U(f,P) \leq \sum_{i=0}^{n}\frac{1}{n}[/tex] which diverges. I am unable to refine the inequality.
 
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  • #2
1.You need to take advantage of the fact that the [itex]a_n[/itex] converge to zero.
2.You don't know that L(f,P) is going to be 0 since the [itex]a_n[/itex] can be less than zero.

Hint:
If you know that for n>N, a_n < 1/k can you produce an upper bound for U(f,P) as the norm of the partition goes to zero?
 
Last edited:
  • #3
a_n is a strictly decreasing sequence converging to 0, a_n is therefore never less than 0.

I think I have an idea on how to proceed, regardless.
 
  • #4
Treadstone 71 said:
a_n is a strictly decreasing sequence converging to 0, a_n is therefore never less than 0.

<opens mouth, switches feet>
Oh, I missed that. It makes things a bit more convenient.
 

FAQ: Proving that a function is Riemann integrable

1. What is Riemann integrability?

Riemann integrability is a concept in calculus that refers to the ability to find the definite integral of a function over a given interval. It is a measure of how well-behaved a function is on that interval.

2. How do you prove that a function is Riemann integrable?

To prove that a function is Riemann integrable, you need to show that the upper and lower Riemann sums of the function converge to the same value as the partition of the interval becomes finer and finer. This can be done by using the definition of Riemann integrability or by using certain theorems and properties.

3. What is the difference between Riemann integrability and other types of integrability?

Riemann integrability is a specific type of integrability that deals with functions that are continuous or have a finite number of discontinuities on a given interval. Other types of integrability, such as Lebesgue integrability, allow for a wider range of functions, including those with infinite discontinuities.

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

Yes, a function can be Riemann integrable on one interval but not on another. This is because the behavior of a function can vary on different intervals, and Riemann integrability is dependent on the behavior of the function on a specific interval.

5. Why is it important to determine if a function is Riemann integrable?

Determining if a function is Riemann integrable allows us to find the area under the curve of the function, which has many practical applications in fields such as physics, engineering, and economics. It also provides a measure of how well-behaved a function is on a given interval, which can help in understanding the behavior of the function and making predictions.

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