PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 10, Pages 2917–2922 S 0002-9939(99)04872-8 Article electronically published on April 23, 1999 LEVEL SETS OF A TYPICAL C n FUNCTION UDAYAN B. DARJI AND MICHA L MORAYNE (Communicated by Frederick W. Gehring) Abstract. We determine the level set structure of a typical C n function. Introduction Assume that F is a function space complete with respect to some norm. We say that a typical function in F satisfies property P if the subfamily of F consisting of those functions which satisfy P is residual (in the sense of Baire category) in F . If y is in the range of f , we call f −1 ({y})a level set of f . In [2] Bruckner and Garg gave a full description of level sets of a typical function from C[0, 1], the space of real-valued, continuous functions defined on [0, 1] and equipped with the sup norm. Namely, they proved: Theorem 1 (Bruckner and Garg). For a typical function f ∈ C[0, 1] there exists a countable set S f ⊆ (min f, max f ) such that the level set f −1 ({y}) is: 1. a nowhere dense perfect set if y ∈ S f ∪{min f, max f }, 2. a single point if y = min f or max f, and 3. the union of a nowhere dense perfect set and an isolated point of f −1 ({y}) if y ∈ S f . In this paper we give the description of level sets of typical functions in C n [0, 1]. Namely, we prove that a typical f ∈ C 1 [0, 1] is either strictly monotone or f has uncountably many level sets having exactly one accumulation point and all other level sets of f are finite. For a typical function in C n [0, 1] with n ≥ 2 the situation is simple. All level sets are finite. Level sets of a typical C n function Let us now introduce some notation and definitions. The symbols N, Q, R will denote the sets of all positive integers, rationals and reals, respectively. We use |A| to denote the cardinality of set A. The restriction of a function f to a set A will be denoted by f |A. We use λ(A) to denote the Lebesgue measure of A for a Lebesgue measurable subset A of R. Received by the editors May 14, 1997 and, in revised form, December 23, 1997. 1991 Mathematics Subject Classification. Primary 26A21, 26A16. Key words and phrases. Level sets, C n functions, typical function. The second author was supported in part by KBN Grant 2P 301 04 307. This paper was written when the second author was visiting the Department of Mathematics of the University of Louisville, Kentucky, USA. c 1999 American Mathematical Society 2917 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use