On the continuous points of semi-continuous functions

Let (X, d) be a complete metric space and E be a non-empty closed set in X. Suppose that f is a lower semi-continuous functions which maps E into R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}. In this paper, we show that f has continuous points on E. Furthermore, we get the set of all such continuous points denote by Cf(E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_f(E)$$\end{document} is dense on E. It should be pointed out that the same problem was also considered by professor Zaslavski (PanAmer Math J 17:1–10, 2007) by using a totally different argument.