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Math Help - functionnal equation

  1. #1
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    functionnal equation

    Hello !
    I have a problem to solve this question :
    Find all real continuous functions f verifing : f(x+1)=f(x)+f(1/x)
    Have you ever seen this, could you help me please ?
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  2. #2
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    There is something that bothers me. You say that [tex]f[tex] is countinous on the number line. Yet, it is undefined for any negative integer!
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  3. #3
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    Quote Originally Posted by tize
    Hello !
    I have a problem to solve this question :
    Find all real continuous functions f verifing : f(x+1)=f(x)+f(1/x)
    Have you ever seen this, could you help me please ?
    Can you tell us some of background to this equation,
    its context etc?

    RonL
    Last edited by CaptainBlack; August 19th 2006 at 07:47 AM.
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  4. #4
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    I would like but I don't know anything about this function except that it is continuous. Somebody ask me about such a function and I don't know where does it come from ...
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  5. #5
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    Quote Originally Posted by tize
    I would like but I don't know anything about this function except that it is continuous. Somebody ask me about such a function and I don't know where does it come from ...
    What makes it unusual that is a composition of,
    f(1/x) which usually are not solved for in functional equations.
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  6. #6
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    Like PH said, this has a cyclical property to it and try replacing all x's with 1/x. The right side of the equation yields the same thing, showing that f(\frac{1}{x}+1)=f(x+1).
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  7. #7
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    Yes we can already proove that :
    f(0)=f(\Phi^{-1})=f(-\Phi)=0 where \Phi is the Gold number : \frac{1+\sqrt 5}{2}
    and
    f has a limit when x\rightarrow\infty, f\rightarrow f(1)
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