I don't know how to prove the statement that $f$ satisfies Lusin's condition (N) on the set $\left\{ x: 0 \ge D^+f(x) >-\infty \right\}$ in the book Differentiation of Real Functions by Andrew Bruckner p129.
More precisely, let $f:\mathbb{R} \to \mathbb{R}$ be continuous and $D^+f(x)$ the upper right Dini derivative defined by $$D^+f(x) = \limsup_{h \to 0^+} \frac{f(x+h)-f(x)}{h}.$$ Let $B\subseteq \mathbb{R}$ be a Borel set and for some $M>0$, $$ 0 \ge D^+f(x) > -M, \forall x \in B.$$ If $\mathscr{L}(B)=0$, how to prove $$\mathscr{L}(f(B))=0?$$
