Small coverings with smooth functions under the Covering Property Axiom

In the paper we formulate a Covering Property Axiom CPAprism, which holds in the iterated perfect set model, and show that it implies the following facts of which (a) and (b) are the generalizations of results of Steprāns [26]. (a) There exists a family F of less then continuum many C1 functions from R to R such that R2 is covered by functions from F in the sense that for every 〈x, y 〉 ∈ R2 there exists an f ∈ F such that either f(x) = y or f(y) = x. (b) For every Borel function f:R → R there exists a family F of less than continuum many “C1 ” functions (i.e., differentiable func-tions with continuous derivatives, where derivative can be infi-nite) whose graphs cover the graph of f