In an accelerating thin liquid film, it is possible for a location to exist in which the flow field passes from subcritical flow, in which waves affect both upstream and downstream locations, to supercritical flow, in which all waves propagate downstream. At that location, referred to as the critical point, the steady-state equation governing the shape of the film exhibits a removable singularity that sets an internal boundary condition on the flow. This boundary condition provides essential mathematical structure as well as physical insight into the flow problem. In previous papers, Cerro and Scriven (1980) and Kheshgi et al. (1992) postulate that neglected higher-order terms may become important and induce an internal boundary layer in the vicinity of a critical point. In this paper, we reconsider the rapid dip coating analysis of Cerro and Scriven (1980) and demonstrate that no such boundary layer exists through the use of a regular asymptotic expansion. Thus, higher-order terms, when sensibly small based on scaling arguments, may be formally neglected in all regions of the flow, and the mathematical simplifications afforded by the film equation containing the critical point are uniformly valid. Although focused on a specific model problem here, it is likely that the mathematical structure uncovered may be generalized to other thin film flows with critical points.