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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 < ''δ'' over which the Cantor function cumulatively rises more than ''ε''.
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 $\sum\limits_{k=1}^Mc(y_k)-c(x_k)=1$. Yep, it's equal to one. Very surprising indeed !
== Alternative definitions ==
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