Is this riemann integrable on [0,1]?

when x is rational and when x is irrational

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- Apr 22nd 2010, 02:57 PMCrazyCat87Riemann Integrability
Is this riemann integrable on [0,1]?

when x is rational and when x is irrational - Apr 22nd 2010, 03:07 PMDrexel28
- Apr 22nd 2010, 03:24 PMCrazyCat87
I guess I'm having trouble with this material since I'm unsure of what you're asking here....

- Apr 22nd 2010, 04:21 PMDrexel28
- Apr 23rd 2010, 07:42 AMCrazyCat87
I understand that it's at most 1 and 0, I just don't know how to prove this using the definition, which I think is what I'm supposed to do

- Apr 23rd 2010, 11:12 AMDrexel28
The point is this. If you give me

*any*partition of into subintervals, let's just call them , we find that . Why? Well, let's look at something. These intervals are presumably non-degenerate (not one point) and so in any of these intervals there are__both__irrational and rational numbers. So, for__any__subinterval since it must contain an irrational. So, . But, why does this last sum equal one? The way I wrote it should give you a clue. Start with that .