A note on the range of the derivatives of analytic approximations of uniformly continuous functions on c0

A real Banach space X satisfies property (K) (defined in [M. Cepedello, P. Hájek, Analytic approximations of uniformly continuous functions in real Banach spaces, J. Math. Anal. Appl. 256 (2001) 80–98]) if there exists a real-valued function on X which is uniformly (real) analytic and separating. We obtain that every uniformly continuous function f:U→R, where U is an open subset of a separable Banach space X with property (K) and containing c0 (thus X=c0⊕Y for some Banach space Y) can be uniformly approximated by (real) analytic functions g:U→R such that (where is the set of partial derivatives ). Similar statements are obtained for uniformly continuous functions f:U→E with values in a (finite or infinite dimensional) Banach space E. Some consequences of these results are studied.