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# $$f(r) = f( r + \frac{1}{2} )$$

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.

Note by Calvin Lin
3 years ago

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On $$[0,1]$$ 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 : $$f(r)=f(r+1/2)$$. · 3 years ago