Cantor function: Difference between revisions

Revision as of 00:42, 30 December 2018

Revision as of 00:44, 30 December 2018

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===Lack of absolute continuity===

Because the [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0, for any positive ''ε<1'' and ''δ'', there exists a finite sequence of [[pairwise disjoint]] sub-intervals with total length <&nbsp;''δ'' over which the Cantor function cumulatively rises more than ''ε''. If a given sequence did not, all but an arbitrarily short portion of each sub-interval containing a point in the Cantor set could be discarded and a finite number equal-length sub-intervals containing different points in the Cantor set added, iteratively, until the cumulative rise was greater than ''ε''. Note that this is just the idea of proof, not the proof itself.

In fact, to every $\delta>0$ there are finitely many pairwise disjoint intervals $(x_k,y_k)$ ($1\leq k\leq M$) with $\sum\limits_{k=1}^My_k-x_k<\delta$ and $\sum\limits_{k=1}^Mc(y_k)-c(x_k)=1$. Very surprising indeed !