Conway's base 13 function

Conway's base 13 function is a mathematical 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

)

{\displaystyle (a,b)}

, the function

f

{\displaystyle f}

takes every value between

f

(

a

)

{\displaystyle f(a)}

and

f

(

b

)

{\displaystyle f(b)}

— but is not continuous.

Conway's base 13 function is an example of a simple-to-define 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. Conway's creation of the function has been attested to by the mathematician Adebisi Agboola, who reported that Raymond Lickorish had told students in a lecture in 1982 about Conway referencing the function during a discussion about continuity.

Sketch of definition

Every real number

x

{\displaystyle 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 {+, −, .}, 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 3629256 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, +, −, .}.