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, it is a function that satisfies a particular intermediate-value property—on any interval (a,b), the function f passes through every point between f(a) and f(b)—but nevertheless isn't continuous.

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. It is thus discontinuous at every point.

Sketch of definition

• Every real number x can be represented in base 13 in a unique canonical way; such representations use the digits 0-9 plus three additional symbols, say {A,B,C}. For example, as you can compute, the number 54349589 has a base-13 representation of B34C128.

• If instead of {A,B,C}, we judiciously choose the symbols {+,-,·}, something interesting happens: some numbers in base 13 will have representations that look like well-formed decimals in base 10: for example, the number 54349589 has a base-13 representation of -34.128. Of course, most numbers won't be intelligible in this way; for example the number 3629265 has the base-13 representation 9+0--7.

• Conway's base 13 function takes in a real number x and considers its base 13 representation as a sequence of symbols { 0, 1, ..., 9, +, -, · }. If from some position onward, the representation looks like a well-formed decimal number r, then f(x)=r. Otherwise, f(x)=0. (Well-formed means that it starts with a + or - symbol, contains exactly one decimal-dot symbol, and otherwise contains only the digits 0-9). For example, if a number x has the representation 8++2.19+0--7+3.141592653..., then f(x) = +3.141592653... .

Definition

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

For example,

$f(12345A3C14.159\dots _{13})=f(A3.C14159\dots _{13})=3.14159\dots$,

$f(B1C234_{13})=-1.234$,

and $f(1C234A567_{13})=0$.

Properties

• According to the intermediate value theorem, every continuous real function $f$ satisfies an intermediate-value property: on every interval (a,b), the function $f$ passes through every point between $f(a)$ and $f(b)$. The Conway base 13 function shows that the converse is false: it satisfies the intermediate-value property, but isn't continuous.

• In fact, the Conway base 13 function satisfies a much stronger intermediate-value property—on every interval (a,b), the function $f$ passes through every real number. As a result it satisfies a much stronger discontinuity property— it is discontinuous everywhere.

• To prove that the Conway base 13 function satisfies this stronger intermediate property, let (a,b) be an interval, let c be a point in that interval, and let r be any real number. Create an base-13 encoding of r as follows: starting with the base-10 representation of r, replace the decimal point with C and indicate the sign of r by prepending either an A (if r is positive) or a B (if r is negative) to the beginning. By definition of the Conway base 13 function, the resulting string ${\hat {r}}$ has the property that $f({\hat {r}})=r$. Moreover, any base-13 string that ends in ${\hat {r}}$ will have this property. Thus, if you replace the tail end of c with ${\hat {r}}$, the resulting number will have f(c') = r. By introducing this modification sufficiently far along the tridecimal representation of $c$, you can ensure that the new number $c'$ will still lie in the interval $(a,b)$. This proves that for any number r you choose, in every interval you can find a point $c'$ such that $f(c')=r$.

• The Conway base 13 function is therefore discontinuous everywhere: a real function that is continuous at x must be locally bounded at x, i.e. it must be bounded on some interval around x. But as shown above, the Conway base 13 function is unbounded on every interval around every point; therefore it is not continuous anywhere.

References

• Oman, Greg (2014). "The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond" (PDF). Missouri J. Math. Sci. 26 (2): 134–150. Archived (PDF) from the original on 2016-08-20.

See also

• Darboux function