06-21-2026, 08:50 AM
Conway's base 13 function
by Wikipedia
Summary
Conway’s base 13 function is a remarkable mathematical construction devised by John Horton Conway that serves as a counterexample to the converse of the Intermediate Value Theorem. The function is defined by interpreting the base-13 expansion of a real number using the symbols 0–9 together with three special symbols that act like “+”, “−”, and “.”; whenever a valid decimal number appears in the tail of the expansion, the function outputs that decimal value, otherwise it outputs 0.
Despite its simple definition, the function has extraordinary properties: on every interval of real numbers it attains every possible real value, making it an everywhere-surjective (and therefore everywhere discontinuous) function whose graph is dense in the plane. It demonstrates that having the intermediate-value property does not imply continuity, making it one of the most famous examples of a “pathological” function in real analysis.
ARTICLE
by Wikipedia
Summary
Conway’s base 13 function is a remarkable mathematical construction devised by John Horton Conway that serves as a counterexample to the converse of the Intermediate Value Theorem. The function is defined by interpreting the base-13 expansion of a real number using the symbols 0–9 together with three special symbols that act like “+”, “−”, and “.”; whenever a valid decimal number appears in the tail of the expansion, the function outputs that decimal value, otherwise it outputs 0.
Despite its simple definition, the function has extraordinary properties: on every interval of real numbers it attains every possible real value, making it an everywhere-surjective (and therefore everywhere discontinuous) function whose graph is dense in the plane. It demonstrates that having the intermediate-value property does not imply continuity, making it one of the most famous examples of a “pathological” function in real analysis.
ARTICLE
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