How to prove that f(x) = cx for some constant c?

[tuesdaymorning54]

Prove that if f is a continuous function on all real numbers such that f(x+y) = f(x) + f(y) for all x, y is an element of real numbers and f is continuous, then f(x) = cx for some constant c (an element of real numbers).

So far...We know that f(nx) = nf(x) for n (element of natural numbers)
Proof: f(2x) = f(x+x) = f(x) + f(x) = 2f(x)

Thank you so much for your help

 

[tkhunny]

That is not a proof. That is only a demonstration. You must show it for ALL positive integers. How shall you proceed?