On Inductively Open Real Functions

In this note, given a locally connected topological space X, we characterize those continuous and locally nonconstant real functions on X which are inductively open there. Throughout this note, X denotes a locally connected topological space. Let / be a real function on X. We recall that / is said to be inductively open in X (see [1]) if there exists a set X* Q X such that f{X*) = f(X) and the function /|X.:X*^/(X)isopen. Recently, in [2], as a consequence of a general lower semicontinuity theorem for certain multifunctions, we have Theorem 1 [2, Théorème 2.4]. LetX also be connected. Then any continuous real function f on X, such that int(/_1 (t)) = 0 for every t G] inf f(X), sup f(X)[, is inductively open in X. It is easy to show by means of simple examples that none of the hypotheses of Theorem 1 can be dropped. In particular, this theorem is no longer true if X is disconnected. Indeed, it suffices to take X = [0, l]u]2,3] and /: X —► R defined as follows: Hx) = {X~l tfx6l°'1l' iK ' \x-2 ifxG]2,3]. / cannot be inductively open in X, since, otherwise, it would be open there, being one-to-one. But / is not open in X (for instance, [0,1] is open in X but /([0,1]) is not open in f{X)). The aim of this note is to characterize those continuous and locally nonconstant real functions on X which are inductively open there. We first recall a lemma established in [3]. Lemma 1 [3, Lemma 3.1]. Let S be a topological space, Y a connected subset of S, s0, si two points ofY, g a real function on S. Moreover, assume: (1) so is a local maximum (resp. minimum) point for g; (2) g(s0) < ff(si) (resp. g{s0) > g(si)); (3) g is continuous at every point ofY. Then there exists s* ElY with the following properties: (i) g(s*) = g{s0); (ii) s* is not a local maximum (resp. minimum) point for g; (iii) s* is not a local minimum (resp. maximum) point for g, provided that for every open set Q Ç S, with fi C\Y # 0, there exists seil such that g(s) =£ fif(so). Received by the editors April 19, 1983. 1980 Mathematics Subject Classification. Primary 54C10, 54C30; Secondary 54D05. 1 Supported by M.P.I. © 1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 485 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use