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Math Help - Function on [0,1]

  1. #1
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    Sep 2009
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    Function on [0,1]

    Hello,

    The problem is as follows:

    Function f is such that: f:[0,1] \rightarrow [0,1], f(0)=0, f(1)=1, f is continuous in 0 and in 1 and the following condition is satisfied:
    f(x)=2f(x/2) for all x \in (0,1].

    The question is: Does only function f(x)=x satisfy the conditions above? We may prove that if f is either convex or concave then this is the only solution. But can we do the same without assuming that?

    Thanks for help
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  2. #2
    Super Member General's Avatar
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    f can not be continuous on neither 0 nor 1.
    Do you mean continuous on 0 from right and on 1 from left?
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  3. #3
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    Yes, I meant continuous in 0 from right and in 1 from left
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