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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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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 04 '13 at 17:35
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Here is a related question. – David Mitra Feb 04 '13 at 17:42
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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 $b=a+\frac{1}{2}$
Thomas Andrews
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