Level-continuity of functions and applications

In this paper, we study the concepts of level-continuity and proper local maximum points of functions defined on a topological space X and, on one hand, we establish that, under adequate conditions, f is level-continuous if f is without proper local maximum points and, on the other, we prove that level-convergence and variational convergence (Γ-convergence) of functions are equivalents when the limit function is level-continuous.