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!

Homework Help: Analysis Function Question

  1. Oct 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Suppose f is continous on [0,1] and f(0)=f(1). Let n be a natural number. Prove that there is some number x, such that f(x)=f(x+1/n).


    2. Relevant equations
    The hints says to consider g(x)=f(x)-f(x+1/n)

    3. The attempt at a solution
    I've tried to consider the function g(x), but I haven't gotten anything useful from it. When I've tried various values of n, like 1/2, 1/3...I've noticed that there are repeating terms and I can manipulate the terms a bit to get like g(0)=-g(1/2) for n=1/2 and the like, but I am not sure where to go with this
     
  2. jcsd
  3. Oct 15, 2013 #2
    What's wrong with x =0 and n=1?
     
  4. Oct 15, 2013 #3

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You don't get to choose ##n##. It is chosen for you. In other words, you have to show there is a solution for ANY ##n##.
     
  5. Oct 15, 2013 #4

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If ##g(0) = 0## then you can simply take ##x = 0##. If ##g(0) > 0##, then I claim there must be some other point ##x## such that ##g(x) < 0##, and then you can apply the intermediate value theorem. Can you prove this claim? Hint: consider ##g(0) + g(1/n) + \ldots + g((n-1)/n)##.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Loading...