Interesting problem from my analysis class

Click For Summary
The discussion revolves around proving that for a continuous function f on the interval [0,1] with f(0) = f(1), there exists a horizontal chord of length 1/n. Participants express skepticism about the problem being homework and debate the necessity of differentiability. The problem is considered intuitively plausible, with connections to the intermediate value theorem noted, though the proof details are seen as tedious. Suggestions include assuming f(x) is non-negative to explore further implications. Overall, the problem is recognized as interesting yet challenging.
StonedPanda
Messages
60
Reaction score
0
Let n be a positive integer and suppose f is continuous on [0,1] and f(0) = f(1). Prove that the graph of f has a horizontal chord of length 1/n. In other words, prove there exists x \in [0,(n - 1)/n] such that f(x+1/n) = f(x)
 
Last edited:
Physics news on Phys.org
no one even wants to try?
 
hint: write f(1) - f(0) as a telescoping sum.
 
StonedPanda said:
no one even wants to try?

I would hazard a guess that people are suspicious that this is your homework.
 
Is f differentiable, or just continuous?
 
seems trivial. indeed trivial for all numbers less than or equal to 1, not just 1/n.

after thinking about an actual proof for a few minutes let me rephrase that as "intuitively plausible", rather than "trivial".

it apparently follows from the intermediate value theorem but the details seem tedious, even elusive. cute problem.
 
Last edited:
assume f(x) non negative, then what?
 

Similar threads

Undergrad Trouble with a passage in Zariski & Samuel's Commutative algebra text
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
4K
Undergrad Proving inequality about dimension of quotient vector spaces
  • · Replies 25 ·
Replies
25
Views
3K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
840
Undergrad Seeking an intuition linking definitions of contravariant and covariant
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad How can I test for positive semi-definiteness in matrices?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Localising a non integral domain at a prime
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
2K
MHB What Is the Upper Bound of Groups of Order in Finite Group Theory?
  • · Replies 2 ·
Replies
2
Views
2K
High School Is it invalid to redefine the sgn function in this way in a proof?
  • · Replies 15 ·
Replies
15
Views
5K