Let f be a function defined on all of r that satisfies the
additive condition f(x+y)=f(x)+f(y) for all x, y∈r.
(a) show that f(0)=0 and that f(−x)=−f(x) for all x∈r .
(b) let k=f(1). show that f(n)=kn for all n∈n, and then prove that
f(z)=kz for all z∈z. now, prove that f(r)=kr for any rational
number r.
(c) show that if f is continuous at x=0, then f is continuous at every point
in r and conclude that f(x)=kx for all x∈r. thus, any additive
function that is continuous at x=0 must necessarily be a linear function
through the origin.