I've been trying to do the following question, but I've been unable to make any conclusions at all. If anyone could offer a suggestion, please do! It's appreciated.
Let $E \subset \mathbb{R}$ be a set of Lebesgue measure zero. Show that there exists a function defined on $\mathbb{R}$ which is continuous and increasing everywhere and that fails to be differentiable at each point in $E$.
By outer regularity, there is a nested sequence of open sets $U_n$ so that $E \subseteq U_n$ and $m(U_n) < 2^{-n}$. Let $$f(x) = x + \sum_{n=1}^\infty m(U_n \cap (-\infty, x))$$ To see that $f$ is non-differentiable on $E$, note that $(f(x+h) - f(x))/h \ge 1 + n$ if $h > 0$ and $(x, x+h) \subseteq U_n$.