Darboux like functions
Richard G. Gibson and
Real Anal. Exchange 22(2), 492--533.
This is a survey concerning Darboux like functions.
By "Darboux like" functions we understand the following classes
We consider the relationships between those classes in different
families of functions:
- Ext - extendable functions;
- ACS - almost continuous functions (in the sense of Stallings);
- Conn - connectivity functions;
- D - Darboux functions;
- PC - peripherally continuous functions;
- SCIVP - functions with the Strong Cantor Intermediate Value Propery;
- CIVP - functions with the Cantor Intermediate Value Property;
- WCIVP - functions with the Weak Cantor Intermediate Value Property;
- PB - functions having the property (B).
Moreover, we study the following algebraic operations on Darboux
like real functions:
- - in the class of all real functions of one real variable;
- - in the class of all real functions of n variables;
- - in the Baire class one;
- - in the class of Borel measurable functions;
- - in the class of Lebesgue measurable functions;
- - in the class of Marczewski measurable functions;
- - in the class of Sierpinski-Zygmund functions;
- - in the class of additive functions;
- - in the class of quasi continuous functions.
Finally we consider cardinal functions defined in terms of
algebraic operations on Darboux like functions.
- - compositions;
- - pointwise and uniform limits;
- - sums and products;
- - maxima and minima.
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Last modified August 9, 1997.