FINE CONVERGENCE OF FUNCTIONS AND ITS EFFECTIVIZATION

In this article, we first discuss the Fine continuity and the Fine convergence in relation to the continuous convergence on [0,1). Subsequently, we treat computability and the effective Fine convergence for a sequence of functions with respect to the Fine topology. We prove that the Fine computability does not depend on the choice of an effective separating set and that the limit of a Fine computable sequence of functions under the effective Fine convergence is Fine computable. Finally, we generalize the result of Brattka, which asserts the existence of a Fine computable but not locally uniformly Fine continuous function.