PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 3, March 1988 MEASURABLE DARBOUX FUNCTIONS J. B. BROWN, P. HUMKE AND M. LACZKOVICH (Communicated by R. Daniel Mauldin) ABSTRACT. We investigate how certain Darboux-like properties of real func- tions (including connectivity, almost continuity, and peripheral continuity) are related to each other within certain measurability classes (including the classes of Lebesgue measureable, Borel, and Baire-1 functions). I. Introduction. Consider the following topological properties that functions / from one topological space, X, to another, Y, may have: Conn: / is a connectivity function if it is true that for every connected subset G of X, f[C is a connected subset of X x Y, D: /is Darboux if for every connected subset C of X, f[C] is a con- nected subset of Y. For functions / from the unit interval 2 = [0,1] into the reals R, f is Conn if and only if it has a connected graph. It is obvious that Conn => D, but it was shown in [12] that Conn <* D. On the other hand, Kuratowski and Sierpinski showed in [13] that within the class Bi of functions /: 2 —► R of Baire's first class, Conn o D. This result has been expanded a great deal. It is shown in Theorem 1.1, Chapter II, of Bruckner's book [3] that within 22i, the following properties are also equivalent toD: PC: /: X —* Y is peripherally continuous if for each x E X and each pair of open sets U Ç X and V Ç Y such that x E U and f(x) E V, there exists an open subset W of U, containing x, such that f\bd(W)] C V. Here, bd(W) denotes the boundary of W. For real functions, this reduces to the condition of Young (see [3, (2), p. 9]). SM: f:I—*R satisfies the condition of Sen and Massera if for each xEl, /(*)e lim f(z), lim f(z) n lim f(z), lim f(z) CC: /: 2 —» R is CC if for each number a, the sets {/ < a} and {a < /} have compact components, Received by the editors September 16, 1986 and, in revised form, November 6, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 26A21; Secondary 54C08. Key words and phrases. Darboux, connectivity, Baire's first class. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 603 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use