  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, $p \in \{ 0,
1, 2, \cdots \} \cup \{ \infty \}$, 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 $A^p$.
