We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory--Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.