Suppose f is continuous R to R and that f(x+y)=f(x)+f(y) for all x and y. Prove that f(x)=xf(1).
Hint: Prove the result for all Z and then for all Q.
Any help appreciated.
The first question I ever asked on this forum!
(Long time ago, Look hier).
If a function is continous it must be a linear function thus proving your statement.
Otherwise, the axiom of choice can construct a non-linear solution
(This functional equation is one of my favorite math things I ever learned)