Is the given function Riemann integrable?

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The discussion centers on the Riemann integrability of the Dirichlet function defined as f(x) = 1/q for rational x = p/q and f(x) = 0 for irrational x within the interval [0, 1]. Participants assert that this function is indeed Riemann integrable because its set of discontinuities has Lebesgue measure zero. The confusion arises from the distinction between this specific Dirichlet function and others referenced in Wikipedia, which states that a different version of the Dirichlet function is not Riemann integrable due to being discontinuous everywhere.

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Nusc
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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?
 
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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.
 
morphism said:
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.
 
mathman said:
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.
 
Ditto Hurkyl -- and in any case, this function is actually continuous everywhere except at a countable set (namely the rationals in [0,1]).
 
Nusc said:
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.
 
HallsofIvy said:
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?
 
rbj said:
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?
 

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