Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions

Abstract

We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the so-called Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate some connections of the Lipschitz derivatives defined on normed spaces to the Fr\'{e}chet derivative and relations between little, big and local Lipschitz derivatives (denoted by $\lip f$, $\Lip f$ and $\LLip f$ respectively) in terms of Baire limit functions. In particular, we prove that $\lip f$ is $\mathcal{F}_{\sigma}$-lower, $\Lip f$ is $\mathcal{F}_{\sigma}$-upper, $\LLip f$ is upper semicontinuous. Moreover, for a function $f$ defined on an open or convex subset of a normed space, the upper Baire limit function of functions $\lip f$ and $\Lip f$ are equal to $\LLip f$.