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Maximum-entropy inference and inverse continuity of the numerical range

We study the continuity of the maximum-entropy inference map for two observables in finite dimensions. We prove that the continuity is equivalent to the strong continuity of the set-valued inverse numerical range map. This gives a continuity condition in terms of analytic eigenvalue functions which implies that discontinuities are very rare. It shows also that the continuity of the MaxEnt inference method is independent of the prior state.