 Cantor function

The Cantor function is a continuous nondecreasing map $C:[0,1]\to [0,1]$ associated with the middle-thirds Cantor set. It is constant on each removed open interval and maps the endpoints to zero and one. On the Cantor set, ternary digits $0,2$ become binary digits $0,1$, with endpoint expansions interpreted consistently.

It has derivative zero almost everywhere because the complement of the Cantor set has full Lebesgue measure and $C$ is locally constant there. Nevertheless $C(1)-C(0)=1$. Thus continuity and an almost-everywhere derivative do not alone justify $C(1)-C(0)=\int C^{\prime }$. The missing property is absolute continuity, illustrating why the hypotheses of the fundamental theorem of calculus matter.

The Cantor function, also known as the Cantor staircase function, is a special function that is defined on the interval \([0, 1]\) and is notable for its unique properties. It is constructed using the Cantor set, which is a well-known fractal. ### Properties of the Cantor Function: 1. **Construction**: The Cantor function is typically constructed in conjunction with the Cantor set.