Conway base 13 function

The Conway base 13 function is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, even though Conway's function f is not continuous, if f(a) < f(b) and an arbitrary value x is chosen such that f(a) < x < f(b), a point c lying between a and b can always be found such that f(c) = x.

The Conway base 13 function

Purpose

The Conway base 13 function was created in response to complaints about the standard counterexample to the converse of the intermediate value theorem, namely sin(1/x). This function is only discontinuous at one point (0) and seemed like a cheat to many. Conway's function, on the other hand, is discontinuous at every point.

Definition

The Conway base 13 function is a function $f: (0,1) \to \mathbb{R}$ defined as follows.

If $x \in (0,1)$ expand x as a tredecimal (a "decimal" in base 13) using the symbols 0,1,2,3,4,5,6,7,8,9,$\cdot$,-,+ (avoid + recurring).

Define f(x) = 0 unless the expansion ends with:

$\pm a_1 a_2 \ldots a_n \cdot b_1 b_2 b_3 \ldots$ (Note: Here the symbols "+" and "-" are used as symbols of base 13 decimal expansion, and do not have the usual meaning of the plus and minus sign; the a_is and b_js are restricted to the digits 0,1,2,...,9).

In this case define $f(x) = \pm a_1 a_2 \ldots a_n \cdot b_1 b_2 b_3 \ldots$ read in decimal (here we use the conventional definitions of the "+" and "-" symbols, and "$\cdot$" is interpreted as a decimal point).

Properties

The important thing to note is that the function f defined in this way satisfies the converse to the intermediate value theorem but is continuous nowhere. That is, on any closed interval [a,b] of the real line, f takes on every value between f(a) and f(b). Indeed, f takes on the value of every real number on any closed interval [a,b] where b > a. To see this, note that we can take any number $c \in (a,b)$ and modify the tail end of its base 13 expansion to be of the form $\pm a_1 a_2 \ldots a_n \cdot b_1 b_2 b_3 \ldots$ (the last three dots are not base 13 digits, but just indicate that the expansion continues), and we are free to make the a_i and b_j whatever we want while only slightly altering the value of c. We can do this in such a way that the new number we have created, call it c', still lies in the interval [a,b], but we have made f(c') a real number of our choice. Thus f satisfies the converse to the intermediate value theorem (and then some). Moreover, if f were continuous at some point, f would be locally bounded at this point, which is not the case. Thus f is a spectacular counterexample to the converse of the intermediate value theorem.

References

• Agboola, Adebisi. Lecture. Math CS 120. University of California, Santa Barbara, 17 December 2005.

See also

• Darboux function