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
In 2018, a much simpler function with the property that every open set is mapped onto the full real line, was published by user Aksel Bergfeldt on the community question and answer site Mathematics Stack Exchange.
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.
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.