2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62185We extend Painlevé's determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general 'multiple-valued' Cauchy's problems. We study $C^0-$continuability (near singularities) of solutions.3 pages, LaTeX; completely revised versionComplex VariablesMathematical Physicstext