The Coulomb-breakup method to extract the cross section for neutron radiative capture at astrophysical energies is analyzed in detail. In particular, its sensitivity to the description of the neutron-core continuum is ascertained. We consider the case of 14 C(n, γ) 15 C for which both the radiative capture at low energy and the Coulomb breakup of 15 C into 14 C+n on Pb at 68 MeV/nucleon have been measured with accuracy. We confirm the direct proportionality of the cross section for both reactions to the square of the asymptotic normalization constant of 15 C observed by Summers and Nunes [Phys. Rev. C 78, 011601 (2008)], but we also show that the 14 C-n continuum plays a significant role in the calculations. Fortunately, the method proposed by Summers and Nunes can be improved to absorb that continuum dependence. We show that a more precise radiative-capture cross section can be extracted selecting the breakup data at forward angles and low 14 C-n relative energies.