\begin{displaymath}
 g(x) = \left\{
 \begin{array}{lr}
 x, & x \in \mathbb{Q}\cap \langle -\infty, 0 \rangle\\
 \frac{x^2}{2}, & x \in [0, \sqrt{2}] \\
 f(x), & x\in \mathbb{Q}\cap \langle \sqrt{2}, +\infty \rangle
 \end{array}
 \right
\end{displaymath}