I am reading some lecture notes and came across the Cantor Function. However, I have some questions about it after reading.
Recall the Cantor set $C \subseteq [0, 1]$ is compact, has Hausdorff dimension $\alpha = \frac{\ln 2}{\ln 3}$ and $H^\alpha$ denotes the Hausdorff measure of dimension $\alpha$. We define the Cantor function as: $$ f(x) = \frac{H^\alpha(C \cap [0, x])}{H^\alpha(C)}. $$ Intuitively, this is the fraction of the Cantor set that lies to the left of $x$. Now define the function $$ g(x) = \inf\{ y: f(y) = x \}. $$ It can be observes that $f(g(x)) = x$ as $f$ is increasing and continuous and thus the infimum is achieved. The lecture note then claims $g$ is not continuous and I would like to understand the reasoning behind it.
In particular, I would like to understand why: $$ g(\frac{1}{2}) = \frac{1}{3} $$ and $$ g(y) \geq \frac{2}{3} $$ for any $y > \frac{1}{2}$.
Hint. Here is a picture of $f$.
In particular, you need to know $f(x) = \frac12$ for all $x \in [\frac13,\frac23]$.