Darboux property for functions of several variables

C. J. Neugebauer
1963 Transactions of the American Mathematical Society  
Introduction. Let En be the n-dimensional Euclidean space. A function /: E1 -ȣ! is said to have the Darboux property if for a < b and u between/(a), f(b) there is te(a,b) satisfying f(t) = u. There are several important classes of functions which possess the Darboux property : approximately continuous functions and ordinary or approximate derivatives. The purpose of this paper is to extend these results to En by introducing a Darboux notion for E" with the view of obtaining the following
more » ... tions : (1) approximately continuous functions on E", (2) derivatives of certain interval functions, and (3) partial derivatives (approximate or ordinary) of linearly continuous functions possess the Darboux property. The Darboux property of a function on Et is equivalent with the notion of a connected mapping. The problem then is to define a class of connected sets in E", called Darboux class, relative to which the Darboux property of a function is defined. The Darboux class for E" should contain the connected open subsets of E" and has to be sufficiently large so as to reduce to the Darboux notion for n -1, and sufficiently small so as to obtain the propositions (1), (2), and (3). With this in mind the extreme classes, i.e., the class of connected open sets and the class of all connected sets, do not form suitable Darboux classes. For the first class there would be no agreement for n = 1 [5] and, since an approximately continuous function on En (n > 1) need not be a connected mapping, the second class is too large.
doi:10.1090/s0002-9947-1963-0148811-6 fatcat:y6vqvxiv45bydon5piyiht2dcm