Is the given function Riemann integrable?

[Nusc]

Here is the classic Dirichlet function:

Let, for x ∈ [0, 1],
f (x) =1 /q if x = p /q, p,q in Z

or 0 if x is irrational.

Show that f (x) is Riemann integrable and give the value of the integral. Is this actually true?

 

[Nusc]

Wikipedia says it's not,

http://en.wikipedia.org/wiki/Lebesgue_integral

 

[morphism]

Your function isn't defined at zero, but let's ignore that. Where does Wikipedia say that it's not integrable? It is in fact Riemann integrable. Its set of points of discontinuity has measure zero.

 

[mathman]

Your function isn't defined at zero, but let's ignore that. Where does Wikipedia say that it's not integrable? It is in fact Riemann integrable. Its set of points of discontinuity has measure zero.

It is Lebesgue integrable, but not Riemann. For Riemann, you need continuity except at a countable set.

 

[Hurkyl]

It is Lebesgue integrable, but not Riemann. For Riemann, you need continuity except at a countable set.

That's a sufficient condition, not a necessary condition. The necessary and sufficient condition is that the function be discontinuous on a set of (Lesbegue) measure zero. (This function satisfies that condition)

I think the problem isn't too hard if you just start writing down Riemann sums, and use approximations to simplify things.

 

[morphism]

Ditto Hurkyl -- and in any case, this function is actually continuous everywhere except at a countable set (namely the rationals in [0,1]).

 

[HallsofIvy]

Wikipedia says it's not,

http://en.wikipedia.org/wiki/Lebesgue_integral

What Wikipedia says is

For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but it does not have a Riemann integral.

That is NOT the "Dirichlet function" the OP was talking about. The particular Dirichlet function Wikipedia is referring to (There are several) is discontinuous everywhere.

 

[rbj]

What Wikipedia says is

For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but it does not have a Riemann integral.

That is NOT the "Dirichlet function" the OP was talking about. The particular Dirichlet function Wikipedia is referring to (There are several) is discontinuous everywhere.

but for x > 0, the given function is always less than the Dirichlet function (where not zero, the given function 1/q is less than 1). (it appears to be periodic with any period that is 1/q for integer q.) and we know that the Dirichlet function has Lebesque integral of zero and if this is Riemann integrable, i think the two integrals (over the same limits) has to be the same, no?

 

[morphism]

but for x > 0, the given function is always less than the Dirichlet function (where not zero, the given function 1/q is less than 1). (it appears to be periodic with any period that is 1/q for integer q.) and we know that the Dirichlet function has Lebesque integral of zero and if this is Riemann integrable, i think the two integrals (over the same limits) has to be the same, no?

I don't see what you're objecting to. Are you arguing that the given function isn't Riemann integrable?