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.© BrainMass Inc. brainmass.com March 4, 2021, 6:45 pm ad1c9bdddf
Let's check some properties of the additive function f(x)
(a) f(x+y)=f(x)+f(y). This is definition.
Because f(x)=f(x-y+y)=f(x-y)+f(y), then f(x-y)=f(x)-f(y)
Because f(x)=f(x+0)=f(x)+f(0), then f(0)=0.
Because f(0)=f(x-x)=f(x)+f(-x)=0, then f(-x)=f(x)
Because f(x)=f(n*(x/n))=n*f(x/n) by (e), then f(x/n)=f(x)/n
This is induced by (e) ...
Properties of additive functions are investigated.