# Riemann Integrable

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?

Last edited:

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

HallsofIvy
Homework Helper
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