Because the [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0, for any positive ''ε<1''<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 $\delta''δ''>0$ there are finitely many pairwise disjoint intervals $(x_k''x<SUB>k</SUB>'',y_k''y<SUB>k</SUB>'')$ ($''1\leq ''≤''k\leq ''≤''M$'') 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>. Yep, it's equal to one. Very surprising indeed !
