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 takes every value between f(a) and f(b)—but is not continuous.

In 2018, a much simpler function with the property that every open set is mapped onto the full real line was constructed by Aksel Bergfeldt.^[1] This function is also nowhere 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.^[2] 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, the number 54349589 has a base-13 representation B34C128.
• If instead of {A, B, C}, we judiciously choose the symbols {+, −, .}, 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 will not 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-point 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 ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ 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:

• ${\displaystyle f(\mathrm {12345A3C14.159} \dots _{13})=f(\mathrm {A3C14.159} \dots _{13})=3.14159\dots ,}$
• ${\displaystyle f(\mathrm {B1C234} _{13})=-1.234,}$
• ${\displaystyle f(\mathrm {1C234A567} _{13})=0.}$

Properties

• According to the intermediate-value theorem, every continuous real function ${\displaystyle f}$ has the intermediate-value property: on every interval (a, b), the function ${\displaystyle f}$ passes through every point between ${\displaystyle f(a)}$ and ${\displaystyle f(b).}$ The Conway base-13 function shows that the converse is false: it satisfies the intermediate-value property, but is not continuous.
• In fact, the Conway base-13 function satisfies a much stronger intermediate-value property—on every interval (a, b), the function ${\displaystyle f}$ passes through every real number. As a result, it satisfies a much stronger discontinuity property— it is discontinuous everywhere.
• From the above follows even more regarding the discontinuity of the function - its graph is dense in ${\displaystyle \mathbb {R} ^{2}}$.
• 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 a 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 ${\displaystyle {\hat {r}}}$ has the property that ${\displaystyle f({\hat {r}})=r.}$ Moreover, any base-13 string that ends in ${\displaystyle {\hat {r}}}$ will have this property. Thus, if we replace the tail end of c with ${\displaystyle {\hat {r}},}$ the resulting number will have f(c') = r. By introducing this modification sufficiently far along the tridecimal representation of ${\displaystyle c,}$ you can ensure that the new number ${\displaystyle c'}$ will still lie in the interval ${\displaystyle (a,b).}$ This proves that for any number r, in every interval we can find a point ${\displaystyle c'}$ such that ${\displaystyle f(c')=r.}$
• 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.
• The Conway base-13 function maps almost all the reals in any interval to 0.^[3]

See also

• Darboux function – All derivatives have the intermediate value propertyPages displaying short descriptions of redirect targets

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.