# 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 following is Conway's own notation.)

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

If $x \in \mathbb{R}$, write $x$ as a tridecimal (a "decimal" in base 13) using the 13 underlined "digit" symbols $\underline{0}$, $\underline{1}$, $\underline{2}$, ..., $\underline{8}$, $\underline{9}$, $\underline{+}$, $\underline{-}$, $\underline{\cdot}$; there should be no trailing $\underline{\cdot}$ recurring. There may be a leading sign, and somewhere there will be a tridecimal point to distinguish 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.)
If from some point onwards, the tridecimal expansion of $x$ consists of an underlined signed ordinary decimal number, $r$ say, then define $f(x) = r$, otherwise define $f(x) = 0$. For example,
$f(\underline{7{+}{\cdot}1}\,.\,\underline{4{+}3{\cdot}14159\ldots}) = f(\underline{7{+}{\cdot}14{+}3{\cdot}141}\,.\,\underline{59\ldots}) = \pi$

Note that the tridecimal point and earlier occurrences of $\underline{+}$ and $\underline{\cdot}$ are ignored, as there are later occurrences of non-decimal digits. (More precisely, to have the $f(x)=r$ case, the trailing part must consist of either $\underline{+}$ or $\underline{-}$, followed by some finite number (possibly zero) of underlined decimal digits, followed by $\underline{\cdot}$, followed by some number (possibly infinitely many) of underlined decimal digits. Other possible cases can be permitted, but it makes no difference to the crucial properties of the function.)

### 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 $\underline{r}$ (that is, $r$ underlined, with $r$ being written as a signed decimal), 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

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