On the continuity and limiting values of functions

Abstract

of the limting values of / as the independent variable £ approaches x and we can also define the continuity of / at some of the points xdX. These topological structures on X will be such that for a noncountable set of points xdX the limiting values of / as £—>x and the continuity of / at x can be interpreted in at least two different ways. For example if X is the real line one an speak about the limiting values of / as £ approaches x from the left and from the right, and of limj^.I_o/(£) and lim{_I+o/(£), and of the left and right continuity of / at x. The exact definition of these topological structures on X is given later. We shall prove two general theorems for such functions: One of these concern the equality of the limits lim/(£) as £ approaches the same point £ under different conditions. Two cases can be distinguished according as lim/(£) is uniquely determined for each of these topological structures or not. The other theorem states that except for a negligible subset of X the function / is continuous with respect to both or neither of the topologies defined on X. These theorems include many old and new results as special cases from the theory of functions of one and of several real variables. For better understanding it is perhaps more suitable to discuss first some of the known results and to state the new ones afterwards. For real valued functions of one real variable some precise results of this type have been known for a long time. The common source of one group of these results is the following theorem due