Cantor function: Difference between revisions

Revision as of 21:20, 23 February 2025

Revision as of 23:45, 24 February 2025

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Equivalently, if <math>\mathcal{C}</math> is the [[Cantor set]] on [0,1], then the Cantor function <math>c:[0,1]\to[0,1]</math> can be defined as

:<math display="block">c(x) =\begin{cases}

\displaystyle \sum_{n=1}^\infty \frac{a_n}{2^n}, & x = \sum_{n=1}^\infty

& \displaystyle \text{if } x = \sum_{n=1}^\infty

\frac{2a_n}{3^n}\in\mathcal{C}\ \mathrm{for}\ a_n\in\{0,1\};

\\ \sup_frac{y\leq x,\, y2a_n}{3^n}\in\mathcal{C}\ \text{for}\ c(y), & xa_n\in [\{0,1]\smallsetminus \mathcal{C}. \end{cases};

\\

\displaystyle \sup_{y\leq x,\, y\in\mathcal{C}} c(y),

& \displaystyle \text{if } x\in [0,1] \setminus \mathcal{C}.

\end{cases}

</math>