Eberlein theorem and norm continuity of pointwise continuous mappings into function spaces

For a pseudocompact (strongly pseudocompact) space T we show that every strongly bounded (bounded) subset A of the space C(T) of all continuous functions on T has compact closure with respect to the pointwise convergence topology. This generalization of the Eberlein-Grothendieck theorem allows us to prove that, for any strongly pseudocompact spaces T, there exist many points of norm continuity for any pointwise continuous, C(T)-valued mapping h, defined on a Baire space X, which is homeomorphic to a dense Borel subset of a pseudocompact space. In particular, this is so, if X is pseudocompact. In the case when T is pseudocompact the same “norm-continuity phenomenon” has place for every strongly pseudocompact space X or, for every Baire space X which is homeomorphic to a Borel subset of some countably compact space.