Let f:[0,1] -> [0,1] be defined as follows:

f(x)=1/n if x=m/n where n,m are integers, n is not zero, and m/n is irreducible.

f(x)= 0 if x is irrational.

Prove that f(x) is Riemann integrable on [0,1]

please help me on this.

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- January 1st 2009, 11:31 PMKat-Manalysis problem
Let f:[0,1] -> [0,1] be defined as follows:

f(x)=1/n if x=m/n where n,m are integers, n is not zero, and m/n is irreducible.

f(x)= 0 if x is irrational.

Prove that f(x) is Riemann integrable on [0,1]

please help me on this. - January 2nd 2009, 04:53 AMHallsofIvy
What do you have to use? That function, the "modified Dirchlet function" (the Dirichlet function itself is 0 for x rational, 1 for x irrational), can be proved to be continuous exactly on the irrational numbers and 0. That means that it is a bounded function whose set of discontinuities has measure 0. There is a theorem that says such a function is Riemann integrable.

- January 5th 2009, 05:35 PMKat-Mstill having a problem