Smooth extensions of Lipschitzian real functions

Abstract:

In this short note we point out that any Lipschitzian real function $f$ defined in a subset $K$ of a Banach space $E$, with $\overline {{\text {span}}} {\text {(K)}} \ne {\text {E}}$, can be extended to a surjective, open and Lipschitzian real function $g$ on $E$ in such a way that, for every $r \in {\mathbf {R}}$, the set ${g^{ - 1}}(r)$ is arcwise connected. In fact, a more refined result is proved.