Characterization of continuity for real-valued functions in terms of connectedness

We prove that for any topological space X and any function f: X -> R such that Gr(f) is connected and (X x R)\Gr(f) is disconnected it follows that f is continuous. This is a characterization of continuity for real-valued functions on arbitrary connected spaces. We also demonstrate that the real line as the range of the function is the most natural setting for this characterization. In particular, it cannot be extended to multidimensional spaces.