Brouwer's Fixed Point Theorem for Arbitrary Intervals

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Is it possible to prove Brouwer's Fixed Point Theorem (one-dimensional version) for intervals other than [-1,1]-->[-1,1], say [1,2]-->[0,3]? If so, how?
 

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Office_Shredder
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For example, f(x)=x-1 is a map from [1,2] to [0,3] with no fixed point. The condition that the domain and codomain are the same space is very important
 

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