Let (left(tau_{ell}, ell geq 0 ight)) be the inverse of the local time at level 0. Prove

Let \(\left(\tau_{\ell}, \ell \geq 0\right)\) be the inverse of the local time at level 0. Prove that, if \(u \in \mathcal{Z}\), then \(u=\tau_{s}\) or \(u=\tau_{s-}\) for some \(s\).

if \(u \in \mathcal{Z}\), either \(L_{u+\epsilon}-L_{u}>0\) for every \(\epsilon\), and \(u=\tau_{s}\) for \(s=L_{u}\), or \(L\) is constant and \(u=\tau_{s-}\) for \(s=L_{u}\).