Assume $u:R^n_{++} \to R$ is a arbitrary continuous function. Does there exists some strictly increasing $h: R \to R$ such that $h \circ u$ is differentiable? I don't think so, but I have not yet managed to find a counterexample. I am not even sure how to approach a "proof" of an counterexample, as you would have to show that for any strictly increasing h, the composition is not differentiable.
This is no exercise I was given, I am asking myself this question out of curiosity.
Let $g = (g_1, g_2): \mathbb [0,1] \to \mathbb [0,1]^2$ be continuous and surjective (a space-filling curve). Suppose $h_1$ and $h_2$ are strictly increasing functions from $[0,1]$ onto $[0,1]$ such that $h_1 \circ g_1$ and $h_2 \circ g_2$ are differentiable. Then $(h_1 \circ g_1, h_2 \circ g_2)$ is a differentiable space-filling curve. But that is impossible. We conclude that for at least one of $g_1$ and $g_2$, no such $h$ exists.