A “Bouquet” of Discontinuous Functions for Beginners in Mathematical Analysis Giacomo Drago, Pier Domenico Lamberti, and Paolo Toni Abstract. We present a selection of a few discontinuous functions and we discuss some peda- gogical advantages of using such functions in order to illustrate some basic concepts of math- ematical analysis to beginners. 1. INTRODUCTION. In this paper we present a selection of several discontinuous real-valued functions of one real variable which we believe could be proposed to any beginner in mathematical analysis, even to students of secondary school dealing with the first notions of calculus. Some of them are elementary and well known, others a bit more sophisticated. Most of them are modeled on the Dirichlet function. Our aim is to point out some pedagogical advantages of using discontinuous func- tions rather than classical analytic functions. Usually, young students tend to think of basic mathematical analysis as a set of rules of calculus and, surely, many under- graduate students would claim to feel more acquainted with the notion of derivative rather than with the notion of function. This is not surprising at all, if we consider that the historical evolution of the notion of function has been very long and troubled; see Youschkevitch [15] and Kleiner [8, 9]. See also Nicholas [11], Deal [3], and Thurston [13] for an interesting discussion about some pedagogical issues related to the definition of function. A modern definition of function can be found, e.g., in the classic book of Bourbaki [2], published more than two hundred years after the defini- tion of Johann Bernoulli (1718). As is pointed out in [15, p. 79], a significant step in this process was the formulation of A. Cournot (1841), which we report here for the convenience of the reader: We understand that a quantity may depend on another [quantity], even in case the nature of this dependence is such that it cannot be expressed in terms of a combination of algebraic symbols. This level of generality is commonly attributed to Dirichlet, who in 1829 proposed his celebrated function D defined on (0, 1) as follows: D(x ) =  1, if x ∈ (0, 1) ∩ Q, 0, if x ∈ (0, 1) \ Q. (1) In fact, this example opened a door to a new world: functions are not just formulas, or analytic expressions, as was commonly assumed in the 18th century. Functions can be defined by very general laws. From a pedagogical point of view, deciding the level of generality of functions to use with young students is not straightforward. According to Kleiner [9, pp. 187–188], it is possible to teach an elementary model of analysis by placing emphasis solely on curves and the equations that represent them, without necessarily talking about func- tions. Kleiner argues that students would find curves more natural than functions and http://dx.doi.org/10.4169/amer.math.monthly.118.09.799 November 2011] A “BOUQUET” OF DISCONTINUOUS FUNCTIONS 799