Problem
Suppose that $f:(a,b)\to\Bbb R$ is monotonic, and $D=\left\{\,x\;\big|\;f\textrm{ is not differentiable at }x\,\right\}$. Try to prove that for each $\eta>0$, $D$ could be covered by a collection (at most countable) of open intervals $\{O_n\}_{n=1}^\infty$ whose total length is smaller than $\eta$.
Motivation
I heard that it's a well-known theorem in measure theory. I wonder whether we could work without measure theory, where our tools are so elementary, just as in calculus course.
Thoughts
Let $\varphi(x)=\limsup_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}-\liminf_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$, then $D=\left\{\,x\;\big|\;\varphi(x)>0\,\right\}$. Let $D_\varepsilon=\left\{\,x\;\big|\;\varphi(x)\ge\varepsilon\,\right\}$, we have $D=\bigcup_{n=1}^\infty D_{1/n}$, therefore it suffices to prove that $D_\varepsilon$ could be covered by a collection (at most countable) of open intervals whose total length is less than $\eta$, for each $\varepsilon,\eta>0$.