Assume that F is continuous on $[a, b]$. Show that $D^{+}(F)(x) = \lim \sup_{h \rightarrow 0+} \frac{F(x + h)-F(x)}{h}$
is measurable.
I need help solving this problem. My guess is trying to use Dini numbers to prove that $D^{+}(F)(x)<\infty$ almost everywhere...I'm not sure if that's the right track but would that even mean is it necessarily measurable?
EDIT: I know there is another proof of this problem on stackoverflow. But, I found a proof for non-decreasing functions that proves in a different way, so I would not like to use that other proof's method, but my other way instead. But, can someone provide a proof for F is only a decreasing function, please?