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Because the [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0, for any positive ''ε'' 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''. 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''.
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}^
== Alternative definitions ==
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