Finitness Principles for Smooth Selection

In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, and for $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 $X = \mathbb{R}^n$.