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    Continuous Functions

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    Suppose that f(x) satisfies the functional equation

    f(x + y) = f(x) + f(y)

    for all x,y in R (the real numbers). Prove that if f(x) is continuous that f(x) = cx where c is a constant. What can you say about f(x) if it is allowed to be discontinuous?

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    Solution Preview

    Let x=0.
    f(0) = f(0) + f(0)
    so f(0)=0

    Since f is continuous
    lim h->0 f(x+h) = lim h->0 f(x-h)

    because of the functional equation,this means

    lim h->0 f(h) = lim h->0 limf(-h)=0

    Also f(2x -x) = f(2x) + f(-x)
    means f(2x) = f(x) - f(-x)

    and f(2x) = f(x) + ...

    Solution Summary

    The notion of discontinuity in relation to a given function is investigated.