# 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

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.