# Absolute Value Integral Proof

katia11

## Homework Statement

Prove that, if f is continuous on [a,b] and

ab= l f(x) l dx = zero

then f(x) = 0 for all x in [a ,b].

## Homework Equations

Hint- from book-
Section 2.4 Exercise 50
Let f and g be continous at c. Prove that if:
(a) f(c) > o, then there exists delta > o such that f(x) > 0 for all x E (c- delta, c + delta)
(b) f(c) < o, then there exists delta > o such that f(x) < 0 for all x E (c- delta, c + delta)
(c) f(c) < g(c) , then there exists delta > o such that f(x) < g(x) for all x E (c- delta, c + delta)

## The Attempt at a Solution

I understand what we are trying to prove, I can visualize it, but I have NO idea what the "hint" has to do anything. I'm really not a "math person" I discovered. That is a terrible realization. I'm not just looking for the answer though, I just have no idea where to start and proofs scare me.

Homework Helper

## Homework Statement

Prove that, if f is continuous on [a,b] and

ab= l f(x) l dx = zero

then f(x) = 0 for all x in [a ,b].

## Homework Equations

Hint- from book-
Section 2.4 Exercise 50
Let f and g be continous at c. Prove that if:
(a) f(c) > o, then there exists delta > o such that f(x) > 0 for all x E (c- delta, c + delta)
(b) f(c) < o, then there exists delta > o such that f(x) < 0 for all x E (c- delta, c + delta)
(c) f(c) < g(c) , then there exists delta > o such that f(x) < g(x) for all x E (c- delta, c + delta)

## The Attempt at a Solution

I understand what we are trying to prove, I can visualize it, but I have NO idea what the "hint" has to do anything. I'm really not a "math person" I discovered. That is a terrible realization. I'm not just looking for the answer though, I just have no idea where to start and proofs scare me.

The hint tells you that:
• If f(c) < 0. Then f(x) < 0, for x close enough to c.
• If f(c) > 0. Then f(x) > 0, for x close enough to c.

Can you prove these two hints? These two are very useful when dealing with continuous functions.

Well, when tackling some problem with so little information like this, one should think right about: Proof by Contradiction. There's a small property (theorem) that you should know:
$$\mbox{If } f \mbox{ is continuous, then } |f| \mbox{ is also continuous.}$$​

The theorem above should be easy to prove using delta-epsilon method. Let's see if you can get it.

Ok, so back to the main problem. As I said earlier, you should use Proof by Contradiction. I'll give you a little push.

Assume that: $$\exists c \in [a; b] : |f(c)| \neq 0$$, so, what can you say about the value of |f(x)|, when x is close enough to c?