Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs (N, L), established in [2], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem [1]. Dynamical thickening leads to a conceptually simple and short proof.472641646FAPESP [2013/20912-4]FAEPEX [1135/2013]CNPq, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico - BrasilFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)