If supremum=infimum, is f Riemann integrable?

1. Jul 29, 2010

GVAR717

This is my first time posting & I am not familiar with how to get all the correct math symbols or how to use Latex, so I did the best I could.

1. The problem statement, all variables and given/known data

Suppose f is bounded on [a,b] and there is a partition P* of [a,b] for which S(f,P*)=S(f,P*). Is f Riemann integrable on [a,b]?

2. Relevant equations

S(f)=sup{S(f,P*): P* is partition of [a,b]}
S(f)=inf{S(f,P*); P* is partition of [a,b]}

3. The attempt at a solution

I know, by a theorem, that S(f)>S(f). I am trying to figure out how to show S(f)<S(f) so that I can say S(f)=S(f). I thought about choosing another partition Pe such that S(f,Pe)-S(f,Pe) would equal some epsilon value, but I don't know what value I should use or where to go next.

If this is the wrong process for this proof, I would love a hint on where to start.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited: Jul 30, 2010
2. Jul 29, 2010

losiu99

Did you mean "for which S(f, P*)=S(f)"? If so, hint: consider characteristic function of rationals on [0,1].

3. Jul 30, 2010

GVAR717

No, I am trying to use a theorem that says a function is Riemann integrable if the supremum equals the infimimum: S(f)=S(f)

4. Jul 30, 2010

losiu99

Are you using Darboux definition of Riemann integral, I mean:
$$\underline{S}(f, P^*)=\sum m_i \Delta x_i$$
$$\overline{S}(f, P^*)=\sum M_i \Delta x_i$$
where
$$m_i = \inf \{f(x): x_{i-1}\leq x \leq x_i\}$$
$$M_i = \sup \{f(x): x_{i-1}\leq x \leq x_i\}$$
or the other popular one, where
$$S(f, P^*)=\sum f(t_i)\Delta x_i$$
for $$x_{i-1}\leq t_i \leq x_i$$
?
Sorry for such question, but I'm confused. Statement like "S(f, P*)=S(f, P*)" suggests the first (Darboux) definition, while "Relevant Equations" section suggests the second (at least for me). Or maybe it's yet another definition? Again, sorry to ask, but when I find out which definition you are using, I think I will be able to help.

5. Jul 30, 2010

GVAR717

It's the first one with the m's.

By the way, how do you get the math symbols? I don't have Latex.

6. Jul 30, 2010

GVAR717

OMG. I just realized that I was making this problem harder than it is.

I can just say that [tex]\underline{S}(f, P^*)-[tex]\overline{S}(f, P^*)=0, which is less than epsilon.

7. Jul 31, 2010

losiu99

Yes, precisely. Well done As for the math symbols, Latex is an inbuild board feature, you don't need to have it on your computer. Just use [ tex ] [ /tex ] tags around your Latex input. If you're unsure about that, ask or quote my post to see the raw input of my previous messages.