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Thread: Another problem (additive function)

  1. #1
    Nov 2006

    Another problem (additive function)

    Suppose f is continuous R to R and that f(x+y)=f(x)+f(y) for all x and y. Prove that f(x)=xf(1).
    Hint: Prove the result for all Z and then for all Q.

    Any help appreciated.
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  2. #2
    Global Moderator

    Nov 2005
    New York City
    The first question I ever asked on this forum!
    (Long time ago, Look hier).

    If a function is continous it must be a linear function thus proving your statement.

    Otherwise, the axiom of choice can construct a non-linear solution

    (This functional equation is one of my favorite math things I ever learned)
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