If $f:I\subseteq \mathbb{R}\to \mathbb{R}$ is a Lebesgue-measurable function ($I$ is a closed interval), $\mu $ is the Lebesgue measure, my question is whether the limit

$\underset{u\to {u}_{0}^{-}}{lim}\mu ({f}^{-1}((u,{u}_{0}]))=0.$

I don't really know how to justify the value of this limit, if it is 0 or possibly $\mu ({f}^{-1}(\{{u}_{0}\}))$. I would appreciate your help.

$\underset{u\to {u}_{0}^{-}}{lim}\mu ({f}^{-1}((u,{u}_{0}]))=0.$

I don't really know how to justify the value of this limit, if it is 0 or possibly $\mu ({f}^{-1}(\{{u}_{0}\}))$. I would appreciate your help.