Discontinuous Birkhoff theorem - Petrenko, O.; Protasov, I.V. (Український математичний вісник, 2007)
A topological space X is called totally recurrent if every mapping f : X → X has a recurrent point. We prove that a Hausdorff space X is totally recurrent if and only if X is either finite or a one-point compactification of an infinite discrete space.