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.