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, even though Conway's function f is not continuous, if f(a) < f(b) and an arbitrary value x is chosen such that f(a) < x < f(b), a point c lying between a and b can always be found such that f(c) = x. In fact, this function is even stronger than this: it takes on every real value in each interval on the real line.
The Conway base 13 function
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. It is thus discontinuous at every point.
(The following is Conway's own notation.)
The Conway base 13 function is a function defined as follows.
- If , write as a tredecimal (a "decimal" in base 13) using the 13 underlined 'digit' symbols , , , ..., , , , , (and avoid recurring). (There may be a leading sign, and somewhere there will be a tredecimal point to distinguish the integer part from the fractional part; these should both be ignored in the sequel.)
- If from some point onwards, the tredecimal expansion of x consists of an underlined signed ordinary decimal number, r say, then define , otherwise . For example,
Note that the tredecimal point and earlier underlined + and are ignored, as there are later occurrences of non-decimal digits. (More precisely, to have the case, the trailing part must consist of either or , followed by some finite number (possibly zero) of underlined decimal digits, followed by , followed by some number (possibly infinitely many) underlined decimal digits. Other possible cases can be permitted, but it makes no difference to the crucial properties of the function.)
The important thing to note is that the function defined in this way satisfies the conclusion of the intermediate value theorem but is continuous nowhere. That is, on any closed interval of the real line, takes on every value between and . Indeed takes on the value of every real number on any closed interval where . To see this, note that we can take any number and modify the tail end of its base 13 expansion to be of the form (with c being written as a signed decimal). We can do this in such a way that the new number we have created, call it , still lies in the interval (by modifying c far enough along its terdecimal representation), but we have made a real number of our choice. Thus satisfies the conclusion of the intermediate value theorem (and then some). Moreover, if were continuous at some point, would be locally bounded at this point, which is not the case. Thus is a spectacular counterexample to the converse of the intermediate value theorem.
- Agboola, Adebisi. Lecture. Math CS 120. University of California, Santa Barbara, 17 December 2005.
- Conway, John H. Explanation given at Canada/USA Mathcamp, 25 July 2013.