Homework Help: Proving a Function is Riemann Integrable

1. Jan 21, 2008

SNOOTCHIEBOOCHEE

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

Let f, g : [a, b] $$\rightarrow$$ R be integrable on [a, b]. Then, prove that h(x) = max{f(x), g(x)} for
x $$\in$$ [a, b] is integrable.
1

2. Relevant equations

Definition of integrability: for each epsilon greater than zero there exists a partition P so that U(f,P)-L(f,P)<epsilon

3. The attempt at a solution

Ok i have absolutley no clue how to do this one. The following graph is how i think the function would look : http://i31.photobucket.com/albums/c373/SNOOTCHIEBOOCHEE/Graph2.jpg

Sorry about the crude drawing, but the very light blue would be h(x)

But i honest to god cant see a way to make U(f,P)-L(f,P)<epsilon a true statement

2. Jan 21, 2008

morphism

Hint: Prove that if f is integrable then so is |f|.

Why does this help?

3. Jan 21, 2008

SNOOTCHIEBOOCHEE

I know that that is a theorem in the book, but i dont see how its applicable here.

i also know that |Integral(f)| < Integral(|f|) comes from that statement.

Last edited: Jan 21, 2008
4. Jan 21, 2008

morphism

Try writing max(f,g) in terms of absolute values. To get a feel for this, try to find such an expression for max(f,0).

5. Jan 21, 2008

SNOOTCHIEBOOCHEE

how do you mean in absolute values?

like max(f,g)<max(|f|,|g|)?

edit: guess not

im trying to figure out this for h(x)= max(f,0)

so we know h(x) = f if f is positive and 0 else.

but i cant figure this out for the other thing.

6. Jan 21, 2008

morphism

max(f,0) = (|f| + f)/2

7. Jan 21, 2008

SNOOTCHIEBOOCHEE

I see where this argument is going. Basically you are gunna describe h(x) as a sum of absolute values of functions we already know are integrable, therefore the result is integrable.

Now to figure out max(f,g)...

8. Jan 21, 2008

SNOOTCHIEBOOCHEE

sorry for the tripple post. but i got a little bit farther

max(f,g)= [(g+|g|)+ (f+|f|)] /2 - min(f,g)

but then i gotta figure out how to describe the min function...

9. Jan 21, 2008

morphism

Try to think about f-g and f+g.

10. Jan 21, 2008

SNOOTCHIEBOOCHEE

ok ya im not getting this

[(g+|g|)+ (f+|f|)] /2 = f+g if f and g are the same sign

11. Jan 21, 2008

morphism

I really don't know how to give you any more hints that don't involve giving you the answer! Maybe try thinking about why (|f|+f)/2 gives us max(f,0). Try to sleep on it.