Proving that if $f$ is continuous and $f(x+y)=f(x)+f(y)$, then $f$ is linear

Question

I'm having some trouble proving the following proposition my teacher asked us to prove:

Let $E$ be a normed space and $f:E\to \mathbb C$ a continuous function such that, for all $x,y\in E$, $$f(x+y)=f(x)+f(y)$$Prove that $f$ is linear.

I was able to prove that $f(0) = 0$ and using induction that $f(qx)=qf(x)$, for any $q\in \mathbb Q$. But I don't know how to prove this in its general form nor do I know what role continuity plays in all of this. How can this be done?

The title is different of the problem in the post? So $f$ is continuous is a hypotheses or not? — A. P., Dec 23 '22 at 15:41
I'll chance the title. the continuity of $f $ is indeed in the hypotheses @A.P. — Eduardo Magalhães, Dec 23 '22 at 15:42
Real/complex numbers are limits of rational ones (standard density theorem). Since you know the result for rationals, continuity implies it for all reals/complex — peek-a-boo, Dec 23 '22 at 15:43
Does this answer your question? I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$ — Robert Israel, Dec 23 '22 at 15:58

score 1 · Accepted Answer · answered Dec 23 '22 at 15:51

You know that $f(qx) = q f(x)$ for any $q \in \mathbb{Q}, x \in E$. To prove that $f(kx) = k f(x)$ for any $k \in \mathbb{R}$, take any sequence of rational numbers $q_1, q_2, \ldots$ with limit $k$. Then $\lim_{n \to \infty} q_n x = kx$, so $\lim_{n \to \infty} f(q_n x) = f(kx)$ because $f$ is continuous. Furthermore, you've already proved that $f(q_n x) = q_n f(x)$, and $\lim_{n \to \infty} q_n f(x) = k f(x)$.

Proving that if $f$ is continuous and $f(x+y)=f(x)+f(y)$, then $f$ is linear

1 Answers1