On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions

Abstract

In this article we prove that if a function $f:X \to {\cal R}$ is the pointwise (discrete) [transfinite] limit of a sequence of real functions $f_n$ with closed graphs defined on complete separable metric space $X$ then $f$ is the pointwise (discrete) [transfinite] limit of a sequence of continuous functions. Moreover we show that each Lebesgue measurable function $f:{\cal R} \to {\cal R}$ is the discrete limit of a sequence of functions with closed graphs in the product topology $T_d\times T_e$, where $T_d$ denotes the density topology and $T_e$ the Euclidean topology.