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 ∂ g ∂c0 (U) ⊂ p>0 p (where ∂ f ∂c0 (U) is the set of partial derivatives {∂ f ∂x (x, y): (x, y) ∈ U}). 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.