Proving an equality exists in a closed segment of a function.

Question

Well , it was hard thinking of a title but ...Anyway

let $f$ be a continuous function.

It is known that $f(0)=f(10)$

Show that there exists a $"c"$ in the segment $[0,5]$ which will satisfy

$f(c) = f(c+5)$

Well, I tried a few things but I don't believe that they quite work ( they don't when I try to intuitively 'sketch' the function so..)

Do you guys have any tips on how to prove it?

Thanks!

Find a function $g$ such that the desired $c$ would be a zero of $g$. Use the given properties of $f$ to argue that $g$ must have a zero. — Daniel Fischer, Apr 05 '14 at 12:47
This might be interesting for you: http://math.stackexchange.com/questions/16374/universal-chord-theorem The post contains also a generalization of this. — Martin Sleziak, Apr 05 '14 at 18:18

score 1 · Accepted Answer · answered Apr 05 '14 at 17:07

Set $g(x) = f(x + 5) - f(x)$. We have $g(0) = f(5) - f(0)$ and $g(5) = f(10) - f(5) = f(0) - f(5) = -g(0)$. Since $g$ is continuous as a difference of continuous functions, and $g$ changes sign between $0$ and $5$, the intermediate value theorem tells us there is some $c \in [0, 5]$ such that $g(c) = 0 \Leftrightarrow f(c + 5) = f(c)$

Proving an equality exists in a closed segment of a function.

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