Cantor Function

Cantor ternary Function

if ${\mathcal {C}}$ is the Cantor set on [0,1], then the Cantor function c : [0,1] → [0,1] can be defined as

$c(x)={\begin{cases}\sum _{n=1}^{\infty }{\frac {a_{n}}{2^{n}}},&x=\sum _{n=1}^{\infty }{\frac {2a_{n}}{3^{n}}}\in {\mathcal {C}}\ \mathrm {for} \ a_{n}\in \{0,1\};\\\sup _{y\leq x,\,y\in {\mathcal {C}}}c(y),&x\in [0,1]\setminus {\mathcal {C}}.\\\end{cases}}$

Properties of Cantor Functions

• Cantor Function is continuous everywhere, zero derivative almost everywhere.
• lack of absolute continuity.
• Monotonicity
• Its value goes from 0 to 1 as its argument reaches from 0 to 1.

Cantor Function Alternative

The Cantor Function can be construct iteratively using homework construction.

References

Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M. (2006). "The Cantor function" (PDF). Expositiones Mathematicae. Elsevier BV. 24 (1): 1–37.