The issue here has to do with the cumulative distribution $c$ being non-decreasing instead of strictly increasing.
This is taken from exercise 5.13 of Chapter I of "Probability and Stochastics" by Cinlar.
It is fairly easy to prove that the functional inverse (or quantile function) $a$ is non-decreasing and right-continuous. Instead, when it comes to showing that
\begin{equation} c(t)=\inf \{u \in \mathbb{R_+} : a(u) > t\}, \; \; \; \; t \in \mathbb{R_+}, \tag{$\ast$} \end{equation}
I am stuck. I know (as it is the following point of this exercise) that \begin{equation} a(c(t)) \geq t \; \; \; \forall t \in \mathbb{R_+} \end{equation} and \begin{equation} a(c(t)) = t \iff c(t+\epsilon)>c(t) \; \; \; \forall \epsilon > 0 \end{equation}
although trying to apply this hasn't helped me thus far. Any help in proving $(\ast)$ would be much appreciated.