  Prox-regularity is a generalization of convexity that includes all C2,
lower-C2, strongly amenable and primal-lower-nice functions. The study of
prox-regular functions provides insight on a broad spectrum of important
functions. Parametrically prox-regular (para-prox-regular) functions are a
further extension of this family, produced by adding a parameter. Such
functions have been shown to play a key role in understanding stability of
minimizers in optimization problems. This document discusses para-prox-regular
functions in Rn: we begin with some basic examples of para-prox-regular
functions and move on to the more complex examples of the convex and nonconvex
proximal average. We develop an alternate representation of a para-prox-regular
function, related to the monotonicity of an f-attentive epsilon-localization as
has been done for prox-regular functions. This extends a result of Levy, who
used an alternate approach to show one implication of the relationship (we
provide a characterization). We analyze two common forms of parametrized
functions that appear in optimization: finite parametrized sum of functions and
finite parametrized max of functions. The example of strongly amenable
functions by Poliquin and Rockafellar is given and a relaxation of its
necessary conditions is presented.
