Extensions of real and vector functions of one variable which preserve differentiability.

Abstract

Let $f: F \to X$ be a locally bounded function from a closed set $F \subset \R$ to a normed linear space. Then there exists its extension $f^* : \R \to X$ which is differentiable at all points at which $f$ is differentiable. Moreover, $f^*$ is Lipschitz if $f$ is Lipschitz and, in the case $X = \R$, the extension ``preserves Dini derivatives''. The paper partly extends results proved by V. Jarn\'\i k (1923), G. Petruska and M. Laczkovich (1974) and J. Ma\v r\'\i k (1984).