Finitness Principles for Smooth Selection

In this paper we prove finiteness principles for $C^{m}\left( $\mathbb{R}$^{n}, $\mathbb{R}$^{D}\right) $-selection,andfor$C^{m-1,1}\left( $\mathbb{R}$^{n}, $\mathbb{R}$^{D}\right) $-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case when the domain is $X={\mathbb{R}}^n$.