One-sided extendability and p-continuous analytic capacities

Using complex methods combined with Baire’s Theorem, we show that one-sided extendability, extendability, and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, \(p\in \{0,1,2, \ldots \}\cup \{\infty \}\), for compact or closed sets in \(\mathbb {C}\). We use these capacities in order to characterize the removability of singularities of functions in the spaces \( A^p \).