Interesting problem from my analysis class

[StonedPanda]

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)

 

[StonedPanda]

no one even wants to try?

 

[StonedPanda]

hint: write f(1) - f(0) as a telescoping sum.

 

[quantumdude]

no one even wants to try?

I would hazard a guess that people are suspicious that this is your homework.

 

[benorin]

Is f differentiable, or just continuous?

 

[mathwonk]

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.

 

[mathwonk]

assume f(x) non negative, then what?