# Properties of additive functions; Bounded; Continuous; Measurable

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

Proof:

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.