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, the function ftakes every value between f and f—but is not continuous.
Purpose
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. It is thus discontinuous at every point.
Every real numberx can be represented in base 13 in a unique canonical way; such representations use the digits 0–9 plus three additional symbols, say. For example, the number 54349589 has a base-13 representation B34C128.
If instead of, we judiciously choose the symbols, something interesting happens: some numbers in base 13 will have representations that look like well-formed decimals in base 10: for example, the number 54349589 has a base-13 representation of −34.128. Of course, most numbers won't be intelligible in this way; for example, the number 3629265 has the base-13 representation 9+0−−7.
Conway's base-13 function takes in a real number x and considers its base-13 representation as a sequence of symbols. If from some position onward, the representation looks like a well-formed decimal numberr, then f = r. Otherwise, f = 0. . For example, if a number x has the representation 8++2.19+0−−7+3.141592653..., then f = +3.141592653....
Definition
The Conway base-13 function is a function defined as follows. Write the argument value as a tridecimal using 13 symbols as "digits": ; 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, and underlined all of the base-13 "digits" to clearly distinguish them from the usual base-10 digits and symbols.
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:
,
,
.
Properties
According to the intermediate-value theorem, every continuous real function has the intermediate-value property: on every interval, the function passes through every point between and. The Conway base-13 function shows that the converse is false: it satisfies the intermediate-value property, but isn't continuous.
In fact, the Conway base-13 function satisfies a much stronger intermediate-value property—on every interval, the function passes through every real number. As a result, it satisfies a much stronger discontinuity property— it is discontinuous everywhere.
To prove that the Conway base-13 function satisfies this stronger intermediate property, let be an interval, let c be a point in that interval, and let r be any real number. Create a base-13 encoding of r as follows: starting with the base-10 representation of r, replace the decimal point with C and indicate the sign of r by prepending either an A or a B to the beginning. By definition of the Conway base-13 function, the resulting string has the property that. Moreover, any base-13 string that ends in will have this property. Thus, if we replace the tail end of c with, the resulting number will have f = r. By introducing this modification sufficiently far along the tridecimal representation of, you can ensure that the new number will still lie in the interval. This proves that for any number r, in every interval we can find a point such that.
The Conway base-13 function is therefore discontinuous everywhere: a real function that is continuous at x must be locally bounded at x, i.e. it must be bounded on some interval around x. But as shown above, the Conway base-13 function is unbounded on every interval around every point; therefore it is not continuous anywhere.
