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We construct a two-sided discontinuous piecewise linear minimal validfunction for the 1-row Gomory--Johnson model which is not extreme, but which isnot a convex combination of other piecewise linear minimal valid functions. Thenew function only admits piecewise microperiodic perturbations. We present analgorithm for computations with a restricted class of such perturbations.

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VI. The curious case of two-sided discontinuous minimal valid functions