# 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 ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ defined as follows.

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

Note that the tridecimal point and earlier occurrences of ${\displaystyle {\underline {+}}}$ and ${\displaystyle {\underline {\cdot }}}$ are ignored, as there are later occurrences of non-decimal digits. (More precisely, to have the ${\displaystyle f(x)=r}$ case, the trailing part must consist of either ${\displaystyle {\underline {+}}}$ or ${\displaystyle {\underline {-}}}$, followed by some finite number (possibly zero) of underlined decimal digits, followed by ${\displaystyle {\underline {\cdot }}}$, 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 ${\displaystyle f}$ defined in this way satisfies the conclusion of the intermediate value theorem but is continuous nowhere. That is, on any closed interval ${\displaystyle [a,b]}$ of the real line, ${\displaystyle f}$ takes on every value between ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$. More strongly, ${\displaystyle f}$ takes as its value every real number somewhere within every open interval ${\displaystyle (a,b)}$.

To prove this, let ${\displaystyle c\in (a,b)}$ and ${\displaystyle r}$ be any real number. Then ${\displaystyle c}$ can have the tail end of its tridecimal representation modified to be ${\displaystyle {\underline {r}}}$ (that is, ${\displaystyle r}$ underlined, with ${\displaystyle r}$ being written as a signed decimal), giving a new number ${\displaystyle c'}$. By introducing this modification sufficiently far along the tridecimal representation of ${\displaystyle c}$, the new number ${\displaystyle c'}$ will still lie in the interval ${\displaystyle (a,b)}$ and will satisfy ${\displaystyle f(c')=r}$.

Thus ${\displaystyle f}$ satisfies a property stronger than the conclusion of the intermediate value theorem. Moreover, if ${\displaystyle f}$ were continuous at some point, ${\displaystyle f}$ would be locally bounded at this point, which is not the case. Thus ${\displaystyle f}$ 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.