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: 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 following is Conway's own notation.)

The Conway base 13 function is a function f:\mathbb{R} \to \mathbb{R} defined as follows.

If  x \in \mathbb{R} , write x as a tridecimal (a "decimal" in base 13) using the 13 underlined "digit" symbols \underline{0}, \underline{1}, \underline{2}, ..., \underline{8}, \underline{9}, \underline{+}, \underline{-}, \underline{\cdot}; there should be no trailing \underline{\cdot} recurring. There may be a leading sign, and somewhere there will be a tridecimal point to distinguish 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.)
If from some point onwards, the tridecimal expansion of x consists of an underlined signed ordinary decimal number, r say, then define f(x) = r , otherwise define f(x) = 0 . For example,
f(\underline{7{+}{\cdot}1}\,.\,\underline{4{+}3{\cdot}14159\ldots}) = f(\underline{7{+}{\cdot}14{+}3{\cdot}141}\,.\,\underline{59\ldots}) = \pi

Note that the tridecimal point and earlier occurrences of \underline{+} and \underline{\cdot} are ignored, as there are later occurrences of non-decimal digits. (More precisely, to have the f(x)=r case, the trailing part must consist of either \underline{+} or \underline{-}, followed by some finite number (possibly zero) of underlined decimal digits, followed by \underline{\cdot}, followed by some number (possibly infinitely many) of underlined decimal digits. Other possible cases can be permitted, but it makes no difference to the crucial properties of the function.)

Properties[edit]

The function f defined in this way satisfies the conclusion of the intermediate value theorem but is continuous nowhere. That is, on any closed interval [a,b] of the real line, f takes on every value between f(a) and f(b). More strongly, f takes as its value every real number somewhere within every open interval (a,b).

To prove this, let c \in (a,b) and r be any real number. Then c can have the tail end of its tridecimal representation modified to be \underline{r} (that is, r underlined, with r being written as a signed decimal), giving a new number c'. By introducing this modification sufficiently far along the tridecimal representation of c, the new number c' will still lie in the interval (a,b) and will satisfy f(c')=r.

Thus f satisfies a property stronger than the conclusion of the intermediate value theorem. Moreover, if f were continuous at some point, f would be locally bounded at this point, which is not the case. Thus f is a spectacular counterexample to the converse of the intermediate value theorem.

References[edit]

  • Agboola, Adebisi. Lecture. Math CS 120. University of California, Santa Barbara, 17 December 2005.

See also[edit]