Convergence near a point and the Arzela-Ascoli-type theorems

Theorems are proved giving necessary and sufficient conditions for the convergence of a sequence of continuous (differentiable) functions to a continuous (differentiable) function. The concepts of convergence near a point and equipotential convergence near a point are introduced. These concepts are introduced locally; on a segment, they are equivalent to the quasiuniform convergence and to the uniform convergence of a sequence of functions, respectively.Translated from Ukrainskii Matemahcheskii Zhurnal, Vol. 45, No. 8, pp. 1090–1095, August, 1993.