# 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, that is, it is an everywhere surjective function[1]. It is thus discontinuous at every point.

### Definition

The Conway base 13 function is a function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ defined as follows. Write the argument ${\displaystyle x}$ 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 ${\displaystyle x}$ is of the form ${\displaystyle Ax_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots }$ where all the digits ${\displaystyle x_{i}}$ and ${\displaystyle y_{j}}$ are in ${\displaystyle \{0,\dots ,9\}}$, then ${\displaystyle f(x)=x_{1}\dots x_{n}.y_{1}y_{2}\dots }$ in usual base 10 notation.
• Similarly, if the tridecimal expansion of ${\displaystyle x}$ ends with ${\displaystyle Bx_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots }$, then ${\displaystyle f(x)=-x_{1}\dots x_{n}.y_{1}y_{2}\dots }$.
• Otherwise, ${\displaystyle f(x)=0}$.

For example,

${\displaystyle f(12345A3C14.159\dots _{13})=f(A3.C14159\dots _{13})=3.14159\dots }$,
${\displaystyle f(B1C234_{13})=-1.234}$,
and ${\displaystyle f(1C234A567_{13})=0}$.

### 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 either ${\displaystyle Ar}$ or ${\displaystyle Br}$ depending on the sign of ${\displaystyle r}$ (replacing the decimal dot with a ${\displaystyle C}$), 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

1. ^ Bernardi, Claudio (February 2016). "Graphs of real functions with pathological behaviors". Soft Computing. 11: 5–6. Retrieved 2018-09-07.