Smooth extensions of Lipschitzian real functions

Abstract

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