Author: Kenneth R. Kellum, Department of Mathematics, San Jose State University, San Jose, California 95192 (kellum@math.sjsu.edu). 
Functions that separate X × R, pp. 1221-1226.
ABSTRACT. Suppose f is a real-valued function defined on a connected topological space X where the complement of the graph of f is disconnected. We prove that f restricted to an open subset of X is continuous and the restriction of f to that open set separates the cross space. Also, we prove that if f has the Gibson property, that is, f(cl(U)) is contained in cl(f(U)) for each open set U, then f is continuous. Other conditions insuring the continuity of f are considered.