# how prove $\exists a,b$ that satisfied in following conditions $0 <a\leq b\leq 1, b-a=\frac12, \text{ and }f(a)=f(b)$ [duplicate]

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Universal Chord Theorem

let $f:[0,1]\mapsto\mathbb R$ be continuous and $f(0)=f(1)$how prove $\exists a,b$ that satisfied in following conditions $$1)0<a\leq b\leq 1$$$$2)b-a=\frac12$$$$f(a)=f(b)$$ thanks in advance

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## marked as duplicate by David Mitra, Micah, Nameless, Jonathan Christensen, rschwiebFeb 4 '13 at 18:47

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The second condition implies $a < b$, contradicting the first condition. Did you mean $0 < a < b \leq 1$ instead of the first condition? – gt6989b Feb 4 '13 at 17:35
Here is a related question. – David Mitra Feb 4 '13 at 17:42

## 1 Answer

Hint: Define $g(x)=f(x+\frac{1}{2})-f(x)$ for $x\in [0,\frac{1}{2}]$.

Then show that $g(a)=0$ for some $a\in(0,\frac{1}{2}]$. Then let $b=a+\frac{1}{2}$

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thanks sir your hint help me to prove it – Maisam Hedyelloo Feb 4 '13 at 18:24