SCHWEITZER 2008 Problem 8

Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$ ? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.)

(translated by Miklós Maróti)