Dirichlet function

[asdf60]

the modified dirichlet function (1/q if x is rational = p/q, 0 if x is irrational) is integrable. How is it integrable? What is the upper step function?

 

[Hurkyl]

That one's even Riemann integrable, if memory serves.

Start with this question: for how many values of x is f(x) nonzero? (Infinitely many, I know that. :tongue2: Be more specific!)

 

[asdf60]

Yes, it is reimann integrable. Well after a bit of thinking, i figured it out. The trick is that for any m, there are finite number of Xs such that f(x) < 1/m. So define the step functions at those points as 1, and the rest as 1/m. But you can make the step functions width arbitrarily small at the points f(x) > 1/m, so they don't contribute to the step function integral. Then obviously, the infemum goes to 0.