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, it is a function that satisfies a particular intermediate-value property—on any interval (ab), the function f takes every value between f(a) and f(b)—but is not continuous.

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, that is, it is an everywhere surjective function.[1] It is thus discontinuous at every point.

Sketch of definition[edit]

  • Every real number x can be represented in base 13 in a unique canonical way; such representations use the digits 0–9 plus three additional symbols, say {A, B, C}. For example, the number 54349589 has a base-13 representation B34C128.
  • If instead of {A, B, C}, 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 {0, 1, ..., 9, +, −, .}. If from some position onward, the representation looks like a well-formed decimal number r, then f(x) = r. Otherwise, f(x) = 0. (Well-formed means that it starts with a + or − symbol, contains exactly one decimal-point symbol, and otherwise contains only the digits 0–9). For example, if a number x has the representation 8++2.19+0−−7+3.141592653..., then f(x) = +3.141592653....

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, 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[edit]

  • According to the intermediate-value theorem, every continuous real function has the intermediate-value property: on every interval (ab), 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 (ab), 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 (ab) 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 (if r is positive) or a B (if r is negative) 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(c') = 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.

See also[edit]

References[edit]

  1. ^ Bernardi, Claudio (February 2016). "Graphs of real functions with pathological behaviors". Soft Computing. 11: 5–6. arXiv:1602.07555. Bibcode:2016arXiv160207555B.