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

Analysis Function Question

  • Thread starter lepton123
  • Start date
  • #1
11
0

Homework Statement


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).


Homework Equations


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

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
 

Answers and Replies

  • #2
761
13
What's wrong with x =0 and n=1?
 
  • #3
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,394
180
What's wrong with x =0 and n=1?
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##.
 
  • #4
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,394
180
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)##.
 

Related Threads on Analysis Function Question

Replies
9
Views
3K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
0
Views
885
Replies
1
Views
1K
Replies
1
Views
398
Top