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Abstract and Applied Analysis
Volume 2011, Article ID 857278, 9 pages
Research Article

Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

Department of Mathematics, SUN, 81100 Caserta, Italy

Received 20 June 2011; Accepted 23 August 2011

Academic Editor: Ljubisa Kocinac

Copyright © 2011 Agata Caserta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology.