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\( f(x) = x \)

Suppose that \( f: [0,1] \rightarrow [0,1] \) is a continuous function. Show that is has a fixed point, i.e there is a real number \(x\) such that \( f(x) = x \).

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
2 years, 7 months ago

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On \([0,1]\), consider : \(g(x)=f(x)-x\), then : \(g(0)=f(0)\geq 0\), and \(g(1)=f(1)-1\leq 0\), because : \(0\leq f(x)\leq 1\).

And note that \(g\) is continuous since it is the sum of two contisinuous functions. Therefore it has a root in \([0,1]\) by the IVT. Which is equivalent with saying that there is some \(x\in [0,1]\) such that \(x=f(x)\). Haroun Meghaichi · 2 years, 7 months ago

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