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Math Help - f(x)=x

  1. #1
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    f(x)=x

    iIf f:[0,1] \to [0,1] is an injection which satisfies f(2x-f(x))=x then f(x)=x for x \in [0,1]
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    If f:[0,1] \to [0,1] is an injection which satisfies f(2x-f(x))=x then f(x)=x for x \in [0,1]
    Proof by contradiction: Suppose that f(x)\ne x for some x, say f(x) = x+\delta where \delta\ne0. Then x = f(2x-f(x)) = f(x-(f(x)-x)) = f(x-\delta), so that f(x-\delta) = (x-\delta) + \delta.

    Now repeat the same argument with x-\delta in place of x, to see inductively that f(x-n\delta) =  (x-n\delta) + \delta for n=1,2,3,... . But for n large enough, x-n\delta will be negative (if \delta>0), or greater than 1 (if \delta<0). In either case that gives a contradiction.
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