This article relies largely or entirely on a
single source. (June 2016) |
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 takes every value between and — but is not continuous.
In 2018, a much simpler function with the property that every open set is mapped onto the full real line was published by Aksel Bergfeldt on the mathematics StackExchange. [1] This function is also nowhere continuous.
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. [2] It is thus discontinuous at every point.
B34C128
.−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
.8++2.19+0−−7+3.141592653...
, then f(x) = +3.141592653....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.
For example:
This article relies largely or entirely on a
single source. (June 2016) |
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 takes every value between and — but is not continuous.
In 2018, a much simpler function with the property that every open set is mapped onto the full real line was published by Aksel Bergfeldt on the mathematics StackExchange. [1] This function is also nowhere continuous.
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. [2] It is thus discontinuous at every point.
B34C128
.−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
.8++2.19+0−−7+3.141592653...
, then f(x) = +3.141592653....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.
For example: