Recall that Sard's theorem says that if $U$ is an open subset of $\mathbb{R}^m$ and $f:U\rightarrow \mathbb{R}^n$ is a $C^k$ function, then the set of critical values has emasure zero if $k>m/n-1$.
My question is the following - Is there a continuously differentiable function $f:I=[0,1]\rightarrow \mathbb{R}$ (that is, a function in $C^1[0,1])$ such that the set of critical values is exactly the standard ternary Cantor set (obtained by the usual deletion of middle thirds construction)? By Sard's theorem above a $C^2$ function would be impossible to construct, since then $h:I^2\rightarrow \mathbb{R}^2$ given by $h(x,y) = f(x) + f(y)$ would contain the entire interval $[0,2]$ as the set of critical values violating Sard's theorem.