there exist additive functions which are not linear

Example 1.

There exists a function $f:ℝ\to ℝ$ which is additive but not linear.

Proof.

Let $V$ be the infinite dimensional vector space $ℝ$ over thefield $\mathbb{Q}$. Since $1$ and $\sqrt{2}$ are two independent vectors in $V$, we can extend the set $\{1,\sqrt{2}\}$ to a basis $E$ of $V$ (notice that here the axiom of choice is used).

Now we consider a linear function $f:V\to ℝ$ such that $f(1)=1$ while $f(e)=0$ for all $e\in E\setminus \{1\}$. This function is $\mathbb{Q}$-linear (i.e. it is additive on $ℝ$) but it is not $ℝ$-linear because $f(\sqrt{2})=0\ne \sqrt{2}f(1)$.∎