On reflection of continuous functions and random processes having local times

Assuming that the local time $\alpha(t,u)$, $t\in[0,\infty)$, $u\in\mathbf R$, of a real-valued continuous function $X(s)$, $s\in[0,\infty)$, is continuous in the time parameter, we show that
$$ -\min_{0\le s\le t}\min(X(s),0)=\int_{-\infty}^0\mathbf{1}(\alpha(t,v)>0)\,dv, $$
where the function $\int_{-\infty}^01(\alpha(t,v)>0)\,dv$ is the local time for $\xi(s)=\alpha(s,X(s))$. We apply this result to random processes.