Noncontinuous subadditive function

Question

Is there any non-continuous additive function $f(x+y)= f(x)+f(y)$ from $\mathbb R$ to $\mathbb R$?

Answer 1

First, the only continuous additive functions from $\mathbb{R}$ to $\mathbb{R}$ are of the form $x\mapsto \alpha x$, for some $\alpha \in \mathbb{R}$. To see this, let $\alpha=f(1)$ and note that $f(\frac{a}{b})=\frac{a}{b} \alpha$, for all rationals $\frac{a}{b}$. Hence, by continuity, for all reals $f(x)=\alpha x$

An additive map from $\mathbb{R}\to \mathbb{R}$ is a linear transformation between $\mathbb{R} \to \mathbb{R}$ as a vector space over $\mathbb{Q}$.

$\mathbb{R}$ as a $\mathbb{Q}$ vector space has a basis $B$ containing $1$, by the Axiom of Choice. Take another element $x_0\neq 1 \in B$. Now, let $T$ map $1$ to $1$, $x_0$ to $0$, and the other basis elements to themselves.

This will define a linear transformation $T:\mathbb{R}\to \mathbb{R}$. However, since $T(1)=1$, if $T$ is continuous, it has to be the identity map. Since $T$ maps $x_0$ to $0$, this means $T$ is discontinuous.