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. In fact, this function is even stronger than this: it takes on every real value in each interval on the real line.

The Conway base 13 function

Purpose

The Conway base 13 function was created as part of a "produce" activity: in this case, the challenge was to produce a simple-to-understand function which takes on every real value in every interval. It is thus discontinuous at every point.

Definition

The Conway base 13 function is a function $f:{\mathbb {R}}\to {\mathbb {R}}$ defined as follows. Write the argument $x$ value as a tridecimal (a "decimal" in base 13) using 13 symbols as 'digits': 0, 1, ..., 9, A, B, C; there should be no trailing C recurring. There may be a leading sign, and somewhere there will be a tridecimal point to separate the integer part from the fractional part; these should both be ignored in the sequel. These "digits" can be thought of as having the values 0 to 12, respectively; Conway originally used the digits "+", "-" and "." instead of A, B, C, and underlined all of the base 13 'digits' to clearly distinguish them from the usual base 10 digits and symbols.

If from some point onwards, the tridecimal expansion of $x$ is of the form $Ax_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots$ where all the digits $x_{i}$ and $y_{j}$ are in $\{0,\dots ,9\}$ , then $f(x)=x_{1}\dots x_{n}.y_{1}y_{2}\dots$ in usual base 10 notation.
Similarly, if the tridecimal expansion of $x$ ends with $Bx_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots$ , then $f(x)=-x_{1}\dots x_{n}.y_{1}y_{2}\dots$ .
Otherwise, $f(x) = 0$ .

For example,

f(12345A3C14.159\dots _{13})=f(A3.C14159\dots _{13})=3.14159\dots

f(B1C234_{13})=-1.234

and

f(1C234A567_{13})=0

Properties

The function $f$ defined in this way satisfies the conclusion of 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 $f(b)$ . More strongly, $f$ takes as its value every real number somewhere within every open interval $(a,b)$ .

To prove this, let $c\in (a,b)$ and $r$ be any real number. Then $c$ can have the tail end of its tridecimal representation modified to be either $Ar$ or $Br$ depending on the sign of $r$ (replacing the decimal dot with a $C$ ), giving a new number $c'$ . By introducing this modification sufficiently far along the tridecimal representation of $c$ , the new number $c'$ will still lie in the interval $(a,b)$ and will satisfy $f(c')=r$ .

Thus $f$ satisfies a property stronger than the conclusion of the intermediate value theorem. 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

Oman, Greg (2014). "The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond" (PDF). Missouri J. Math. Sci. 26 (2): 134–150.