  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, $$\sup \left\{
{0.03in} \dim_{H} \left\{ {0.03in} y \ : \ \dim_{H} (f^{-1}(y)) = 1 {0.03in}
\right\} \ : \ f \in C^{k} {0.03in} \right\} = 1,$$ for any $k \in
\mathbb{Z}_{\geq 0}$.
