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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.
[edit] The Conway base 13 function
[edit] Purpose
The Conway base 13 function was created in response to complaints about the standard counterexample to the converse of the intermediate value theorem, namely sin(1/x). This function is only discontinuous at one point (0) and seemed like a cheat to many. Conway's function on the other hand, is discontinuous at every point.
[edit] Definition
The Conway base 13 function is a function 
defined as follows.
- If
expand x as a "decimal" in base 13 using the symbols 0,1,2,...,9,
,-,+ (avoid + recurring). - Define f(x) = 0 unless the expansion ends
(Note: Here the symbols "+", "-" and "." are used as symbols of base 13 decimal expansion, and do not have the usual meaning of the plus sign, minus sign and decimal point).
- In this case define
(here we use the conventional definitions of the "+", "-" and "." symbols).
