Arc-smooth functions and cuspidality of sets

A function f is arc-smooth if the composite f ◦ c with every smooth curve c in its domain of definition is smooth. On open sets in smooth manifolds the arc-smooth functions are precisely the smooth functions by a classical theorem of Boman. Recently, we extended this result to certain tame closed sets (namely, Hölder sets and simple fat subanalytic sets). In this paper we link, in a precise way, the cuspidality of the (boundary of the) set to the loss of regularity, i.e., how many derivatives of f ◦ c are needed in order to determine the derivatives of f. We also discuss how flatness of f ◦ c affects flatness of f. Besides Hölder sets and subanalytic sets we treat sets that are definable in certain polynomially bounded o-minimal expansions of the real field.