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Using complex methods combined with Baire's Theorem we show that one-sidedextendability, extendability and real analyticity are rare phenomena on variousspaces of functions in the topological sense. These considerations led us tointroduce 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 ofsingularities of functions in the spaces $A^p$.

One sided extendability and p-continuous analytic capacities