A continuous monotone function is differentiable almost everywhere by the rising-sun route

Replacing $F$ by $-F$ if necessary, we may assume that $F$ is nondecreasing. For each integer $N\ge 1$, the first inequality of One-sided Hardy-Littlewood inequalities for the Dini derivatives of a continuous monotone function with $u=a$, $v=b$, and $R=N$ shows that the sets $\{x\in [a,b):D^{+}F(x)>N\}$ and, after reflecting the interval, $\{x\in (a,b]:D^{-}F(x)>N\}$ have measure at most $(F(b)-F(a))\mathrm{/}N$. Hence $D^{+}F$ and $D^{-}F$ are finite almost everywhere.

Fix rationals $0<r<R$ and put $E_{r,R}:=\{x\in (a,b):D^{+}F(x)>R\text{ and }{D}_{-}F(x)<r\}\mathrm{.}$ For each rational $h>0$, the maps $x\mapsto \frac{F(x+h)-F(x)}{h}\text{on }(a,b-h),x\mapsto \frac{F(x)-F(x-h)}{h}\text{on }(a+h,b)$ are continuous because $F$ is continuous. Therefore $\{D^{+}F>R\}={\bigcap }_{m\ge 1}{\bigcup }_{\begin{array}{c}h\in \mathbb{Q}\\ 0<h<1\mathrm{/}m\end{array}}\{x\in (a,b-h):\frac{F(x+h)-F(x)}{h}>R\},$ and similarly $\{D_{-}F<r\}={\bigcap }_{m\ge 1}{\bigcup }_{\begin{array}{c}h\in \mathbb{Q}\\ 0<h<1\mathrm{/}m\end{array}}\{x\in (a+h,b):\frac{F(x)-F(x-h)}{h}<r\}\mathrm{.}$ Hence both sets are Borel, so $E_{r,R}$ is Lebesgue measurable. For any subinterval $[u,v]\subseteq [a,b]$, the two inequalities of One-sided Hardy-Littlewood inequalities for the Dini derivatives of a continuous monotone function give $R\lambda (E_{r,R}\cap [u,v])\le F(v)-F(u)$ and $(R-r)\lambda (E_{r,R}\cap [u,v])\le R(v-u)-F(v)+F(u)$. Adding them yields $(2R-r)\lambda (E_{r,R}\cap [u,v])\le R(v-u)$, so $\lambda (E_{r,R}\cap [u,v])\le \frac{R}{2R-r}(v-u)\mathrm{.}$ The constant $\frac{R}{2R-r}$ is strictly less than $1$. Therefore no point of $E_{r,R}$ can be a density-one point of $E_{r,R}$. By the Lebesgue density theorem Lebesgue density theorem, $E_{r,R}$ is null.

Applying step 2.1 to the reflected function $x\mapsto F(a+b-x)$ shows that for every rational $0<r<R$ the set $E_{r,R}^{\mathrm{\prime }}:=\{x\in (a,b):D^{-}F(x)>R\text{ and }{D}_{+}F(x)<r\}$ is null as well. Taking the countable union over rational pairs and using A countable union of measure-zero sets has measure zero, by countable choice, we conclude that outside a null set one has $D^{+}F\le D_{-}F$ and $D^{-}F\le D_{+}F$. Combined with the sidewise inequalities $D_{+}F\le D^{+}F$ and $D_{-}F\le D^{-}F$ from The four Dini derivatives always exist in the extended reals, satisfy the one-sided order inequalities, and detect finite differentiability, this forces $D_{+}F=D^{+}F=D_{-}F=D^{-}F$ at almost every point where the four Dini derivatives are finite.

Steps 1.1 and 3.1 show that all four Dini derivatives agree finitely almost everywhere on $(a,b)$. Therefore The four Dini derivatives always exist in the extended reals, satisfy the one-sided order inequalities, and detect finite differentiability implies that $F^{\mathrm{\prime }}$ exists almost everywhere. Since a null set in the Lebesgue sense is the same as elementary measure zero on the line (A subset of $\mathbb{R}$ has Lebesgue outer measure zero if and only if it has measure zero in the sense of countable closed-interval covers), this is exactly the claimed almost-everywhere statement.