# Riemann Integrability

• Apr 22nd 2010, 01:57 PM
CrazyCat87
Riemann Integrability
Is this riemann integrable on [0,1]?

$f(x)= 0$ when x is rational and $f(x)= 1$ when x is irrational
• Apr 22nd 2010, 02:07 PM
Drexel28
Quote:

Originally Posted by CrazyCat87
Is this riemann integrable on [0,1]?

$f(x)= 0$ when x is rational and $f(x)= 1$ when x is irrational

Is the set of discontinuities of measure zero? What's $\sup_{x\in I}f(x)$ and $\inf_{x\in I}f(x)$ for ANY subinterval $I\subseteq [0,1]$?
• Apr 22nd 2010, 02:24 PM
CrazyCat87
I guess I'm having trouble with this material since I'm unsure of what you're asking here....
• Apr 22nd 2010, 03:21 PM
Drexel28
Quote:

Originally Posted by CrazyCat87
I guess I'm having trouble with this material since I'm unsure of what you're asking here....

What is the biggest and smallest values f takes on any subinterval?
• Apr 23rd 2010, 06:42 AM
CrazyCat87
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, 10:12 AM
Drexel28
Quote:

Originally Posted by CrazyCat87
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

The point is this. If you give me any partition of $[0,1]$ into subintervals, let's just call them $I_1,\cdots,I_n$, we find that $U(P,f)=1$. 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, $\sup_{x\in I_k}f(x)=1$ for any subinterval since it must contain an irrational. So, $U(P,f)=\sum_{j=0}^{n}\sup_{x\in I_j}f(x)\Delta I_j=\sum_{j=1}^{n}\delta I_j=1-0=1$. But, why does this last sum equal one? The way I wrote it should give you a clue. Start with that .