We prove in this paper that if G is a domain in the complex plane satisfying adequate topological or geometrical conditions then there exists a large (dense or closed infinite-dimensional) linear submanifold of boundary-regular holomorphic functions on G all of whose nonzero members are not continuable across any boundary point of G.Ministerio de Ciencia y Tecnología (MCYT). EspañaPlan Andaluz de Investigación (Junta de Andalucía
