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, it is a function that satisfies a particular intermediate-value property—on any interval (''a'', ''b''), the function ''f'' takes every value between ''f''(''a'') and ''f''(''b'')—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.

Sketch of definition

* 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 . 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 will not 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 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

The Conway base-13 function is a function $f: \Reals \to \Reals$ defined as follows. Write the argument $x$ value as a tridecimal (a "decimal" in base 13) 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 $x$ is of the form $A x_1 x_2 \dots x_n C y_1 y_2 \dots$ where all the digits $x_i$ and $y_j$ are in $\,$ then $f(x) = x_1 \dots x_n . y_1 y_2 \dots$ in usual base-10 notation.
* Similarly, if the tridecimal expansion of $x$ ends with $B x_1 x_2 \dots x_n C y_1 y_2 \dots,$ then $f(x) = -x_1 \dots x_n . y_1 y_2 \dots.$
* Otherwise, $f(x) = 0.$

For example:
* $f(\mathrm \dots_) = f(\mathrm \dots_) = 3.14159 \dots,$
* $f(\mathrm_) = -1.234,$
* $f(\mathrm_) = 0.$

Properties

* According to the intermediate-value theorem, every continuous real function $f$ has the intermediate-value property: on every interval (''a'', ''b''), the function $f$ passes through every point between $f(a)$ and $f(b).$ The Conway base-13 function shows that the converse is false: it satisfies the intermediate-value property, but is not continuous.
* In fact, the Conway base-13 function satisfies a much stronger intermediate-value property—on every interval (''a'', ''b''), the function $f$ passes through ''every real number''. As a result, it satisfies a much stronger discontinuity property— it is discontinuous everywhere.
* From the above follows even more regarding the discontinuity of the function - its graph is dense in $\mathbb^2$.
* To prove that the Conway base-13 function satisfies this stronger intermediate property, let (''a'', ''b'') 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 $\hat$ has the property that $f(\hat) = r.$ Moreover, ''any'' base-13 string that ends in $\hat$ will have this property. Thus, if we replace the tail end of ''c'' with $\hat,$ the resulting number will have ''f''(''c'') = ''r''. By introducing this modification sufficiently far along the tridecimal representation of $c,$ you can ensure that the new number $c'$ will still lie in the interval $(a, b).$ This proves that for any number ''r'', in every interval we can find a point $c'$ such that $f(c') = r.$
* 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

*

References

* {{cite journal, last = Oman, first = Greg, title = The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond, journal = Missouri J. Math. Sci., volume = 26, issue = 2, date = 2014, p = 134–150, url = http://www.uccs.edu/Documents/goman/Converse%20of%20IVT.pdf, archive-url=https://web.archive.org/web/20160820112316/https://www.uccs.edu/Documents/goman/Converse%20of%20IVT.pdf , archive-date=2016-08-20, url-status=live

Functions and mappings
John Horton Conway
Special functions