We construct a two-sided discontinuous piecewise linear minimal valid function for the 1-row Gomory--Johnson model which is not extreme, but which is not a convex combination of other piecewise linear minimal valid functions. This anomalous behavior results from combining features of Hildebrand's two-sided discontinuous extreme functions and Basu--Hildebrand--Koeppe's piecewise linear extreme function with irrational breakpoints. The new function only admits piecewise microperiodic perturbations. We present an algorithm for computations with a restricted class of such perturbations.