On some extensions of almost continuous functions and of connectivity functions. (English) Zbl 0797.26006
Real functions \(f\) defined on the entire plane and depending only on the first coordinate, i.e., \(f(x,y)= f(x)\), are considered. As on the real line such functions can be discontinuous but having the Darboux property.
However, in the paper under review it is shown that, in contrast to the situation on the line, any such \(f\) is in fact continuous if we merely know it to be almost continuous or to be a connectivity function.
However, in the paper under review it is shown that, in contrast to the situation on the line, any such \(f\) is in fact continuous if we merely know it to be almost continuous or to be a connectivity function.
Reviewer: B.Kirchheim (Bratislava)
MSC:
26B05 | Continuity and differentiation questions |
54C08 | Weak and generalized continuity |
54C30 | Real-valued functions in general topology |