Exercise 2, § 6

Let $B$ be a base of $\mathbf{R}$ considered as a vector space over the field $\mathbf{Q}$ ("Hamel base"). $B$ is uncountable ($§ 2$, Exercise 2). Let $\varphi$ be a bijection of a subset $C \neq B$ of $B$ onto the set $B$. Define a mapping $f$ of $\mathbf{R}$ into $\mathbf{R}$ as follows: if $x = \sum_{\xi \in B} \lambda(\xi) \xi$, where $\lambda(\xi) \in \mathbf{Q}$ and $\lambda(\xi) = 0$ for all but a finite number of elements $\xi \in B$, then $f(x) = \sum_{\xi \in C} \lambda(\xi) \varphi(\xi)$. Show that $f(x + y) = f(x) + f(y)$, but that, for each $z \in \mathbf{R}$, $f^{-1}(z)$ is a dense subset of $\mathbf{R}$, which implies that $f$ is not bounded above nor below in any interval of $\mathbf{R}$.