TThe note presents a new approach to the notion of continuity of real function at a point. It is applied to obtain a characterization of continuity at a point with respect to *-topology (Hashimoto topology), density topology and $I$-density topology (Wilczyński topology). The latter is closely related to the definition of density point of measurable set formulated by W. Wilczyński in .
"Continuity in Terms of Functional Convergence." Real Anal. Exchange 25 (2) 869 - 878, 1999/2000.