Question: This construction removes the second quarter after each iteration. Picture from Wikipedia:

![Asymmetric Cantor Set](https://upload.wikimedia.org/wikipedia/commons/d/d0/AsymmCantor.png)

[Wikipedia](https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) gives the Hausdorff Dimension as $\log\_2 \phi$ where $\phi$ is the Golden Ratio. 

Intuitively, the dimension tells me that this set, scaled down by a factor of two, will "fit inside of itself" 1.618... times. 

My intuition is leaning on the definition of the "self-similarity" dimension though, which I realize is not the same as the Hausdorff Dimension given by Wikipedia, but I also know that for simple fractal sets like this, the Hausdorff and self-similarity dimensions usually coincide.

In my analysis class last year, we talked briefly about the definition of the Hausdorff-measure and Hausdorff-dimension, but I've found it very difficult to locate examples of people actually showing how to calculate this dimension for any but the most basic objects.

Answer: 
The function is defined $(0,\infty)$, continuous and derivable... Why would not exist?

$$\ln(1) = 0 \rightarrow \lim\_{x \rightarrow 1^{-}}{\ln(x)\cdot \ln(x)} = 0 \cdot 0 = 0 $$
The function is nowhere defined for $x\le1$, so no.