# Dirichlet function upper and lower sum

• Jul 26th 2011, 09:00 AM
CountingPenguins
Dirichlet function upper and lower sum
Question:

Suppose that f(x)=7 if x is rational and f(x)=3 if x is irrational on the interval [1,3]. Let P be a partition of [1,3].

What is the lower sum: I have that the lower sum is 0.

What is the upper sum: I have that the upper sum is 4.

What is sup L(f,P) over all partitions: 7

What is inf U(f,P) over all partitions: 3

And that the function is integrable over [1,3]

Am I on the right track with this?
• Jul 26th 2011, 09:07 AM
FernandoRevilla
Re: Dirichlet function upper and lower sum
Quote:

Originally Posted by CountingPenguins
And that the function is integrable over [1,3]

Lebesgue-integrable, but not Riemann-integrable.
• Jul 26th 2011, 12:01 PM
Also sprach Zarathustra
Re: Dirichlet function upper and lower sum
Quote:

Originally Posted by CountingPenguins
Question:

Suppose that f(x)=7 if x is rational and f(x)=3 if x is irrational on the interval [1,3]. Let P be a partition of [1,3].

What is the lower sum: I have that the lower sum is 0.

What is the upper sum: I have that the upper sum is 4.

What is sup L(f,P) over all partitions: 7

What is inf U(f,P) over all partitions: 3

And that the function is integrable over [1,3]

Am I on the right track with this?

Look here:

http://www.mathhelpforum.com/math-he...tml#post664645