1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Analysis of Continuous functions

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Let f : R → R be continuous on R and assume that P = {x ∈ R : f(x) > 0} is non-empty. Prove that for any x0 ∈ P there exists a neighborhood Vδ(x0) ⊆ P.


    2. Relevant equations



    3. The attempt at a solution
    If you choose some x, y ∈ P, since f(x) is continuous then |f(x) - f(y)| < ε for some ε>0
    then |x-y|<δ for some δ(ε)

    I don't really know where I'm going with this, but I know it has something to do with the question...
    Can someone point me in the right direction?
     
  2. jcsd
  3. Oct 18, 2011 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You have to start with a correct statement of what continuity at x means. Finish this sentence correctly:

    The statement that f(x) is continuous at x means ......
     
  4. Oct 19, 2011 #3
    that for any ε>0 there exists δ(ε) such that for any y, if |x-y|<δ then |f(x)-f(y)|<ε.
    I don't see how this is different from what I put down?
     
  5. Oct 19, 2011 #4
    I'm finding it hard just to put it down into mathematical terms. I can prove it by sketching a drawing, but having trouble translating that...
     
  6. Oct 19, 2011 #5

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That may be why you are having difficulty with this type of problem. But leaving that aside for now, you have a value x0 where f(x0) > 0. The above statement with x = x0 says you can get f(y) close to f(x0). If you can do that is there some way you can guarantee f(y) is positive?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Analysis of Continuous functions
Loading...