Let $\lambda$ be the Lebesgue measure and $F(x)$ be a distribution function (i.e. right continuous and non decreasing with range [0,1]. Show that $$ \lambda\big\{u\in (0,1]: F_l^{\leftarrow}(u) \neq F_r^{\leftarrow}(u)\big\} = 0, $$ where
$F_l^{\leftarrow}(u) = \inf\{t: F(t) \geq u\}$
$F_r^{\leftarrow}(u) = \inf\{t: F(t) > u\}$
I think the proof would be about showing that $F_l^{\leftarrow}(u) = F_r^{\leftarrow}(u)$ for every $u \in (0,1]$, (since $\lambda${$\emptyset$} = $0$), but I do not have any idea about how to proceed. I also tried to draw the function and see the behaviour in case of jump but nothing, this kind of funcion really confuse me.
Any help would be really appreciated. Thanks!