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We investigate three competing notions that generalize the notion of a facetof finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnsonmodel. These notions were known to coincide for continuous piecewise linearfunctions with rational breakpoints. We show that two of the notions, extremefunctions and facets, coincide for the case of continuous piecewise linearfunctions, removing the hypothesis regarding rational breakpoints. We prove anif-and-only-if version of the Gomory-Johnson Facet Theorem. Finally, weseparate the three notions using discontinuous examples.

Facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem