Darboux like functions
by

Richard G. Gibson and Tomasz Natkaniec

Real Anal. Exchange 22(2), 492--533.

This is a survey concerning Darboux like functions. By "Darboux like" functions we understand the following classes 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).

We consider the relationships between those classes in different families of 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.

Moreover, we study the following algebraic operations on Darboux like real functions:

- compositions;
- pointwise and uniform limits;
- sums and products;
- maxima and minima.

Finally we consider cardinal functions defined in terms of algebraic operations on Darboux like functions.

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Last modified August 9, 1997.