On linearization and continuous selections of functions

Abstract

Let f n i = 1,…p, be a finite number of differentiate functions on the Euclidean n-space. We study continuous selections of these functions. The concept of a (non-degenerate) critical point is introduced. In a neighborhood of a noncritical point, the continuous selection turns out to be topologicaily equivalent with a linear function. This result remains true for locally Lipschitz functions. Around a nondegenerate critical point, the continuous selection f is shown to be topologically equivalent with CS(L) +Q, where CS(L) is a specific continuous selection of (k+ 1) linear functions on the Euclidean k-space, and where Q is the sum of (positive and negative) squares of the remaining (n-k) coordinate functions. Finally, the case p≤3 is treated in detail, and we make a link with quasi-differentiate functions.