Convergence near a point and the Arzela-Ascoli-type theorems (Q1339639)

In this note necessary and sufficient conditions for the convergence of a sequence of continuous (differentiable) functions to a continuous (differentiable) function are proved. Concepts of convergence near a point and equipotential convergence near a point are introduced. These concepts are local, on a segment they are equivalent to quasiuniform convergence and to uniform convergence, respectively.