Conway’s Base 13 Function
Conway’s Base 13 Function is a pathological function defined by John Horton Conway that maps every real number to every real number. It serves as a counterexample to the converse of the Intermediate Value Theorem, demonstrating that a function can take every value in an interval without being continuous.
Mathematical Definition
The function relies on the representation of real numbers in base 13. The digits 10, 11, and 12 are typically represented by symbols such as A, B, and C (or similar distinct characters) to distinguish them from the numerical digits 0-9.
- Construction: A real number is mapped to a value by interpreting the tail of its base-13 expansion as a standard base-10 decimal number.
- Condition: If the base-13 expansion of contains a specific pattern (e.g., a sequence starting with a sign, followed by digits, then a decimal point, and more digits), the function extracts the number represented by that sequence.
- Result: The function is Darboux property (has the intermediate value property) but is discontinuous everywhere.
Key Properties
- Surjectivity: The function takes every real value on every interval, no matter how small.
- Discontinuity: It is discontinuous at every point in its domain.
- Non-constructibility: While defined explicitly, it is not computable in the standard sense due to the requirement of inspecting infinite expansions.
Related Concepts
- Pathological Function
- Intermediate Value Theorem
- Base-N Representation
- John Horton Conway