Let [Formula: see text] be a Ck-smoothly (with k≥1) bounded pseudoconvex domain and [Formula: see text] denote its Bergman kernel function. In this article the question is investigated, whether the function [Formula: see text] is continuous up to the boundary in the topology of the extended real line [Formula: see text]. We give two counterexamples: one in the class of finite type domains with k = ∞ and one in the class of convex domains with k = 1.