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.

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.