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We investigate the relation between preimage multiplicity and topological entropy for continuous maps. An argument originated by Misiurewicz and Przytycki shows that if every regular value of a C1 map has at least m preimages then the topological entropy of the map is at least log m. For every integer, there exist continuous maps of the circle with entropy 0 for which every point has at least m preimages. We show that if in addition there is a positive uniform lower bound on the diameter of all pointwise preimage sets, then the entropy is at least log m.