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    Properties of additive functions; Bounded; Continuous; Measurable

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    Let f : R --> R be an additive function i.e. f(x+y) = f(x) + f(y) for all x,y in R.

    1. If f is bounded at a point, then f is continuous at that point.
    2. If f is measurable, then f is linear i.e. f(x) = cx for some c in R.

    I have already proved that f is continuous if and only if f is linear, and I have proven that if f is continuous at a point then it is continuous everywhere.

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    Solution Preview

    Let's check some properties of the additive function f(x)
    (a) f(x+y)=f(x)+f(y). This is definition.
    (b) f(x-y)=f(x)-f(y)
    Because f(x)=f(x-y+y)=f(x-y)+f(y), then f(x-y)=f(x)-f(y)
    (c) f(0)=0
    Because f(x)=f(x+0)=f(x)+f(0), then f(0)=0.
    (d) f(-x)=-f(x)
    Because f(0)=f(x-x)=f(x)+f(-x)=0, then f(-x)=f(x)
    (e) f(nx)=nf(x)
    Because f(nx)=f(x)+f(x)+...+f(x)=nf(x)
    (f) f(x/n)=f(x)/n
    Because f(x)=f(n*(x/n))=n*f(x/n) by (e), then f(x/n)=f(x)/n
    (g) f((m/n)x)=(m/n)*f(x)
    This is induced by (e) ...

    Solution Summary

    Properties of additive functions are investigated.