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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
3 years 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 $$x=f(x)$$. · 3 years ago