  In this paper we settle the two-dimensional case of a conjecture involving
unknown semialgebraic functions with specified smoothness. More precisely, we
prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with
respect to $C^m_{loc}\left( \mathbb{R}^{2},\mathbb{R}^{D}\right) .$ If
$\mathcal{H}$ has a section, then it has a semialgebraic section.
