Conway base 13 function explained

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

, the function takes every value between and - 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 published by Aksel Bergfeldt on the mathematics StackExchange.^[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

can be represented in base 13 in a unique canonical way; such representations use the digits 0–9 plus three additional symbols, say . For example, the number 54349589 has a base-13 representation B34C128.

• If instead of, 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

Notes and References

1. Web site: Open maps which are not continuous . Stack Exchange Mathematics . 2018-09-27 . 2023-07-10 . In an answer to the question.
2. Bernardi. Claudio. Graphs of real functions with pathological behaviors. Soft Computing. February 2016. 11. 5–6. 1602.07555. 2016arXiv160207555B.