Dirichlet function

  • Thread starter asdf60
  • Start date
  • #1
81
0

Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
17
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!)
 
  • #3
81
0
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.
 

Related Threads for: Dirichlet function

  • Last Post
Replies
4
Views
3K
Replies
1
Views
1K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
8K
Replies
4
Views
9K
Replies
4
Views
2K
Top