# Prove that if f and g are integrable on [a, b], then so is fg

1. Feb 24, 2009

### a1010711

3. The attempt at a solution

It suffices to show that f^2 is integrable, since

fg= [(f+g)^2-(f-g)^2]/4

The function x --> x^2 is uniformly continuous on the range of f,

im not sure how to turn this into a formal proof, im lost

Riemann integration

Last edited: Feb 24, 2009
2. Feb 24, 2009

### yyat

Which definition of integrability are you using?

3. Feb 24, 2009

### HallsofIvy

Staff Emeritus
You can't prove it, it's not true. For example, if f(x)= 1 if x is rational, -1 if x is irrational, the f2(x)= 1 so f2(x) is integrable on, say, [0, 1] but f is not. Also, if g(x)= 2, then g2(x)= 4 so is integrable on [0,1] but fg(x)= 2 if x is rational, -1 if x is irrational is not itegrable.

4. Feb 24, 2009

### D H

Staff Emeritus
What if you can find a function f that is integrable on [a,b] but not square integrable?

That much is true.

That's fine, but the title of the thread is "Prove that if f and g are integrable on [a, b], then so is fg", so you are picking some f that violates the given conditions.