Around the Taylor's theorem on convergence of sequences of functions

Taylor's theorem on convergence of sequences of functions is another version of Egoroff's theorem. Egoroff's theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor's theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. In this paper some results concerning Taylor's theorem are presented. A relationship between pointwise convergence, uniform convergence and the Taylor's type of convergence is considered. We investigate properties of sequences of real numbers involved in Taylor's concept of convergence. Two unsolved problems are presented.