Conway base 13 function

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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. 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[edit]

Purpose[edit]

The Conway base 13 function was created as part of a "produce" activity[clarification needed]: 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.

Definition[edit]

The Conway base 13 function is a function defined as follows. Write the argument 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.

  • If from some point onwards, the tridecimal expansion of is of the form where all the digits and are in , then in usual base 10 notation.
  • Similarly, if the tridecimal expansion of ends with , then .
  • Otherwise, .

For example,

,
,
and .

Properties[edit]

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 . More strongly, takes as its value every real number somewhere within every open interval .

To prove this, let and be any real number. Then can have the tail end of its tridecimal representation modified to be either or depending on the sign of (replacing the decimal dot with a ), giving a new number . By introducing this modification sufficiently far along the tridecimal representation of , the new number will still lie in the interval and will satisfy .

Thus satisfies a property stronger than the conclusion of the intermediate value theorem. 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.

References[edit]

See also[edit]