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 ff 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.