Exploring Darboux Functions: The Conway Base 13 Function and Beyond

The existence of a function f: [a, b] rightarrow mathbb{R} that is discontinuous everywhere but satisfies the intermediate value property has fascinated mathematicians since the 19th century. Such functions are known as Darboux functions, named after the French mathematician Jean-Gaston Darboux who first highlighted their properties. This article delves into these intriguing functions, focusing on two key examples: the derivative of a function and the Conway base 13 function. We will explore their definitions, significance, and the available literature for further study.

Introduction to Darboux Functions

In the realm of real analysis, the intermediate value property (IVP) states that if a function takes values f(a) and f(b) at points a and b, then it takes any value between f(a) and f(b) at some point within the interval [a, b]. However, Darboux functions take this property to an extreme, being discontinuous everywhere yet still displaying this behavior.

Derivatives and Darboux Functions

The 19th-century French mathematician Jean-Gaston Darboux made a significant contribution by proving that every derivative has the intermediate value property. This result, while elegant, does not fully answer our question since derivatives are continuous at many points. Darboux illustrated this by constructing a derivative that is discontinuous at numerous points, demonstrating that a Darboux function need not be continuous.

However, while derivatives are discontinuous at some points, they are still continuous at many points, making them unsuitable for our specific requirement of a function that is discontinuous everywhere on the interval [a, b]. This observation arises from the fact that derivatives are well-behaved in many respects, which limits their utility for demonstrating the existence of such extreme functions.

The Conway Base 13 Function

The Conway base 13 function is another fascinating example of a Darboux function that meets our criteria. This function was introduced by the renowned mathematician John Horton Conway and is renowned for its counterintuitive behavior. It is constructed using the base 13 numeral system and exhibits the intermediate value property everywhere on the interval [0, 1], yet it is discontinuous at every rational point. This makes it an excellent candidate for our discussion.

Construction of the Conway Base 13 Function

The Conway base 13 function is defined as follows: Given a real number x in [0, 1], express it in base 13 and replace every occurrence of the digit 0 with the digit 6. The resulting number, which we call y, represents the value of the Conway base 13 function at x. This function is discontinuous at all rational points because the replacement of the digit 0 with 6 results in a dramatic change in the function's value, while it is continuous at all irrational points.

Properties of the Conway Base 13 Function

Despite its discontinuous nature, the Conway base 13 function still satisfies the intermediate value property. To see why, consider any two points a and b in [0, 1] with corresponding function values f(a) and f(b). Since the function's values can be arbitrarily close to each other (due to the base 13 construction), it must take on any value between f(a) and f(b) at some point within the interval [a, b].

Sources for Further Study

To embark on a deeper study of Darboux functions, I recommend consulting the following sources:

Darboux Continuity: AM Bruckner and JG Ceder, ldquo;Darboux Continuity.rdquo; Jber. Deutsch. Math.-Verein. 67 (1964/65), no. Abt. 1, pp. 93-117. This paper provides a detailed survey of Darboux continuity, including simple proofs and references to more advanced literature. Darboux's Theorem: Real Analysis (Wikipedia): This Wikipedia article offers an accessible introduction to Darboux functions, including the Conway base 13 function and other related mathematical concepts.

These resources will provide a comprehensive understanding of the properties and implications of Darboux functions, along with practical examples and theoretical insights.