  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 prove an
if-and-only-if version of the Gomory-Johnson Facet Theorem. Finally, we
separate the three notions using discontinuous examples.
