User:Odinegative/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.

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 "decimal" in base 13 using the symbols 0,1,2,...,9, $\cdot$ ,-,+ (avoid + recurring).

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

:: $\pm a_1 a_2 \ldots a_n . b_1 b_2 b_3 \ldots$ (Note: Here the symbols "+", "-" and "." are used as symbols of base 13 decimal expansion, and do not have the usual meaning of the plus sign, minus sign and decimal point).

:In this case define $f(x) = \pm a_1 a_2 \ldots a_n . b_1 b_2 b_3 \ldots$ (here we use the conventional definitions of the "+", "-" and "." symbols).

=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, $f(x)$ takes on the value of every real number on any closed interval $[a,b]$ . 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 $\pm a_1 a_2 \ldots a_n . b_1 b_2 b_3 \ldots$ , 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 $f(x)$ satisfies the converse to the intermediate value theorem (and then some). However, it is not hard to see, using a similar argument, that $f(x)$ is continuous nowhere. Thus $f(x)$ is a counterexample to the converse of the intermediate value theorem.

References

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