Conway base 13 function
Encyclopedia
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.
, namely sin(1/x). This function is only discontinuous at one point (0) and seemed like a cheat to many. Conway's function, on the other hand, is discontinuous at every point.
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
John H. Conway as a counterexample to the converse of the intermediate value theorem
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
. In other words, even though Conway's function f is not continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
, 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.
Purpose
The Conway base 13 function was created in response to complaints about the standard counterexample to the converse of the intermediate value theoremIntermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
, namely sin(1/x). This function is only discontinuous at one point (0) and seemed like a cheat to many. Conway's function, on the other hand, is discontinuous at every point.
Definition
The Conway base 13 function is a function defined as follows.- If expand as a tredecimal (a "decimal" in base 13Base 13Base-13, tridecimal, tredecimal, or triskadecimal is a positional numeral system with thirteen as its base. It uses 13 different digits for representing numbers...
) using the symbols 0,1,2,3,4,5,6,7,8,9,,-,+ (avoid + recurring). - Define unless the expansion ends with:
- (Note: Here the symbols "+" and "-" are used as symbols of base 13 decimal expansion, and do not have the usual meaning of the plus and minus sign; the s and s are restricted to the digits 0,1,2,...,9).
- In this case define read in decimal (here we use the conventional definitions of the "+" and "-" symbols, and "" is interpreted as a decimal point).