A pathological construction for real functions with large collections of level sets

Consider all the level sets of a real function. We can group these level sets according to their Hausdorff dimensions. We show that the Hausdorff dimension of the collection of all level sets of a given Hausdorff dimension can be arbitrarily close to 1, even if the function is differentiable to some level. By definition of Hausdorff dimension it is clear, for any real function $f(x)$ and any $\alpha \in [0,1]$, that $\dim_{H} \left{ {0.03in} y \ : \ \dim_{H} (f^{-1}(y)) \geq \alpha {0.03in} \right} \leq 1$. What is surprising, and what we show, is that this is actually a sharp bound. That is, $$\sup \{0.03in{\dim }_H\{0.03iny:{\dim }_H(f^{-1}(y))=10.03in\}:f\in C^{k}0.03in\}=1,$$ for any $k \in $\mathbb{Z}$_{\geq 0}$.