Each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$ when $f$ is additive

Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an additive function such that $f(1)=0$. Then for each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$.

Since $f$ is additive and $f(1)=0$ so $f(x+1)=f(x)$. How do we prove $f(I)=f( \mathbb{R})$ ??

Answer 1

Hints:

• It can be shown that $f(q)=0$ for each $q\in\mathbb Q$.
• It can be shown that for every $r\in\mathbb R$ there is a $q\in\mathbb Q$ with $r+q\in I$.