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One of the central questions of Analysis:was: what precisely must be added to pointwise convergence of a sequence of continuous functions to preserve the continuity of the limit? In 1841 Weierstrass discovered that uniform convergence yields continuity of the limit function. In 1983 Arzelà found a necessary and sufficient condition under which the pointwise limit of a sequence of real valued continuous on a compact interval is continuous. In 2009 Caserta, Di Maio, Hola survey a wide range of deep generalizations. In 2010 Caserta , Di Maio, Kocinac  focus on recent directions in this area, e.g.the statistical approach . New notion of convergences  for nets of  partial functions were tackled by  Beer, Caserta, Di Maio, Lucchetti in 2013.