Finely continuously differentiable functions

In Lávička [A remark on fine differentiability, Adv. Appl. Clifford Algebras 17 (2007) 549–554], it is observed that finely continuously differentiable functions on finely open subsets of the plane are just functions which are finely locally extendable to usual continuously differentiable functions on the whole plane. In this note, it is proved that, under a mild additional assumption, this result remains true even in higher dimensions. Here the word “fine” refers to the fine topology of classical potential theory.