  The ground state energy of a finite-dimensional one-parameter Hamiltonian and
the continuity of a maximum-entropy inference map are discussed in the context
of quantum critical phenomena. The domain of the inference map is a convex
compact set in the plane, called the numerical range. We study the differential
geometry of its boundary in relation to the ground state energy. We prove that
discontinuities of the inference map correspond to $C^1$-smooth crossings of
the ground state energy with a higher energy level. Discontinuities may appear
only at $C^1$-smooth points of the boundary of the numerical range considered
as a manifold. Discontinuities exist at all $C^2$-smooth non-analytic boundary
points and are essentially stronger than at analytic points or at points which
are merely $C^1$-smooth (non-exposed points).
