Suppose that \( f: [0,1] \rightarrow [0,1] \) is a continuous function satisfying \( f(0) = f(1) \). Show that there is a real number \( r \) such that \( f(r) = f( r + \frac{1}{2} ) \).

This is a list of Calculus proof based problems that I like. Please avoid posting complete solutions, so that others can work on it.

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TopNewestOn \([0,1]\) define \(g(x)=f(x) - f(x+1/2)\). Note that : \(g(0)=-g(1/2)\) since : \(f(0)=f(1)\).

This means that there is \(r\in [0,1/2]\) such that : \(f(r)=f(r+1/2)\). – Haroun Meghaichi · 3 years, 4 months ago

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