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We give the explicit point $\boxed{x=1/4}$. Repeatedly choose the left remaining third and then the right remaining third. After $2m$ choices, the resulting closed interval is $[L_m,R_m]$, whereThese formulas follow from a finite geometric series. They show $L_m<1/4<R_m$, and that both endpoints tend to $1/4$. The point $1/4$ is also strictly inside each intermediate left third: its distance from $L_m$ is one quarter of the current length. It therefore never enters any assigned closed middle third or its endpoints, so $1/4\in C$.

Endpoint values in the Cantor function construction obey the same left/right choices, with halving in place of thirding. ThusAll these endpoints are in the domain of the partial function: $L_m$ is the right endpoint of a middle third assigned at round $2m$, and $R_m$ is the left endpoint of a middle third assigned at round $2m-1$.

Given $\epsilon >0$, choose $m$ so that $(2/3)4^{-m}<\epsilon$, and put $\delta =9^{-m}/4$. If $t\notin C$ and $|t-1/4|<\delta$, then $L_m<t<R_m$. The assumed monotonicity givesConsequently $|f(t)-1/3|<\epsilon$. This proves directly, on the stated partial domain,There are defined points arbitrarily close to $1/4$, for example $L_m$ and $R_m$, so the limit of a function is not vacuous. No continuous extension has been assumed. The residual $C$ here omits the endpoints that the question assigns values to; it is the usual Cantor set with those assigned endpoints removed.

For $\delta >0$, scale-restricted Hausdorff content is the infimum of ${\sum }_j(\mathrm{diam}{U}_j{)}^s$ over countable covers of $F$ by sets of diameter at most $\delta$, using the convention in Hausdorff measure. It increases as $\delta$ decreases; its limit is Hausdorff measure. Unrestricted Hausdorff content imposes no diameter bound and can behave differently.

Any explicit cover supplies an upper bound on scale-restricted Hausdorff content, rather than its exact value. The level-$n$ intervals of the Cantor set, for example, giveIf this right-hand side tends to zero, then ${\mathcal{H}}^s(C)=0$. If it grows, that alone proves no positive lower bound: a different cover might be cheaper. The mass distribution principle addresses precisely this gap by bounding the cost of every sufficiently small cover.

A Schottky group is a free, discrete group of Möbius transformations constructed by pairing disjoint Jordan curves on the Riemann sphere, taking the exterior of one bounded disk to the interior of its partner. For round circles it is classical. Strictly separated defining disks give a Cantor set as the limit set of a Kleinian group. The tangent-disk extension permits parabolic elements and is often called a noded or T-Schottky group; its Kleinian limit set need not be a Cantor set.