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

Function Monotone on Some Interval

  • Thread starter Icebreaker
  • Start date
  • #1
Icebreaker
I need the following proposition in order to prove another theorem, and I can't seen to find it in my textbook. Any hints on how to proceed, or whether it's actually TRUE, would be helpful.

"If f is defined and continuous on some interval I, then there exists subintervals I'=[x-a,x+b], for some real numbers a and b, at every point x in I such that f is monotone on I'."
 
Last edited by a moderator:

Answers and Replies

  • #2
StatusX
Homework Helper
2,564
1
I don't think that's true. Consider the function:

[tex]f(x)=\left\{\begin{array}{cc}x \mbox{ sin}(\frac{1}{x}),&\mbox{ if } x \neq 0\\0,&\mbox{ if } x=0\end{array}[/tex]

This is continuous, but is not monotone on any interval containing 0.
 
Last edited:
  • #3
Icebreaker
Ok, I've changed my approach. What about:

"If f is defined and continuous on an interval I and c is in I such that f(c)>0, then there exists an interval I' in I where c is in I' such that f(x)>0 for every x in I'."
 
Last edited by a moderator:
  • #4
StatusX
Homework Helper
2,564
1
That's true. Just use the epsilon delta definition of continuity, taking delta as f(c).
 

Related Threads on Function Monotone on Some Interval

Replies
3
Views
786
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
2
Views
981
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
2
Views
723
  • Last Post
Replies
12
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
14
Views
3K
  • Last Post
Replies
15
Views
2K
Top