• Support PF! Buy your school textbooks, materials and every day products Here!

Absolute Value Integral Proof

  • Thread starter katia11
  • Start date
  • #1
18
0

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.
 

Answers and Replies

  • #2
VietDao29
Homework Helper
1,423
2

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:
[tex]\mbox{If } f \mbox{ is continuous, then } |f| \mbox{ is also continuous.}[/tex]​

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

Related Threads on Absolute Value Integral Proof

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
10K
  • Last Post
Replies
2
Views
7K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
978
  • Last Post
Replies
1
Views
785
  • Last Post
Replies
1
Views
747
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
4
Views
1K
Replies
1
Views
2K
Top