RECENT ADVANCES AND OPEN PROBLEMS IN FIRST-RETURN ANALYSIS

摘要

In conclusion, all three of the classes considered here have some right to be called the two-dimensional analog of first-return continuous functions of one variable, i.e, the Baire one, Darboux functions. But because the radially first-return approachable functions preserve the connectivity of line segments (whereas gradients do not) and have continuity properties stronger than Baire one, Darboux functions on the line, perhaps one of the other two classes is more "natural." At any rate, it would now be interesting to completely characterize these three classes, or at least characterize their sets of continuity points.