This is from Tao's blog: https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/comment-page-3/ In proving the monotone differentiation theorem, Tao proved a lemma(lemma $59$ in the note), after which he follows up with Remark $60$: “Note if ${F}$ was not assumed to be continuous, then one would lose a factor of ${C}$ here from the second part of Lemma $56$,and one would then be unable to prevent ${\overline{D^+} F}$ from being up to ${C}$ times as large as ${\underline{D_-} F}$.” I wonder what does this mean and why.
For reference I also put lemma $56$ here, though it’s contained in the link above: Lemma $56$ (One-sided Hardy-Littlewood inequality) Let ${F: [a,b] \rightarrow {\mathbb R}}$ be a continuous monotone non-decreasing function, and let ${\lambda > 0}$. Then we have
$\displaystyle m( \{ x \in [a,b]: \overline{D^+} F(x) \geq \lambda \} ) \leq \frac{F(b)-F(a)}{\lambda}$.
Similarly for the other three Dini derivatives of ${F}$.
If ${F}$ is not assumed to be continuous, then we have the weaker inequality
$\displaystyle m( \{ x \in [a,b]: \overline{D^+} F(x) \geq \lambda \} ) \leq C\frac{F(b)-F(a)}{\lambda}$
for some absolute constant ${C>0}$.