Functions in a Hardy field not ultimately C 

Complex Variables 32:1-6 (1997)

© 1997 OPA (Overseas Publishers Association) Amsterdam B.V.

AMS 1991:12H05, 12J25, 26A12, 26A27, 26E10, 26E30, 34E99

Abstract: I construct a class of functions, whose germs belong to Hardy fields and all of whose derivatives a fortiori ultimately exist, but the functions are not ultimately C . The existence of such functions, while counterintuitive at first glance, is explained by the fact that the higher order derivatives exist in progressively smaller neighborhoods of +. A function not a Hardy field satisfying the required smoothness properties was given in [M. Boshernitzan, An Extension of Hardy's Class L of "Orders of Infinity", J. d'Analyse Math 39:235-255 (1981)]. I provide a proof of the required smoothness properties of this function (omitted by Boshernitzan), and then use this function in the present construction.

@article{dg:ncinf,author={Gokhman, D.},
title={Functions in a Hardy field not ultimately C-infinity},
journal={Complex Variables},volume={32},pages={1--6},year={1997}}