Cantor function: Difference between revisions

Revision as of 17:23, 16 January 2019

Revision as of 01:14, 17 January 2019

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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''&nbsp;<&nbsp;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 &nbsp;''ε''.

In fact, to every ''δ''&nbsp;>&nbsp;0 there are finitely many pairwise disjoint intervals (''x<SUB>k</SUB>'',''y<SUB>k</SUB>'') (''1''&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''M'') with <math>\sum\limits_{k=1}^My_k-x_k<\delta</math> and <math>\sum\limits_{k=1}^Mc(y_k)-c(x_k)=1</math>.

== Alternative definitions ==