Fix an $ a \in \mathbb{R}_{>0} \backslash \ \left \{\tfrac{2M+1}{2N+1} : M,N \in \mathbb{N}\right \} $, and define $ f: \mathbb{N} \to \mathbb{R} $ by: \begin{equation} f(m) = m + \lfloor a m - \tfrac{a}{2} + \tfrac12 \rfloor. \end{equation}
Because of the constraint on a, I claim that $ f $ has a left inverse, given by:
\begin{equation}\label{key} f^{-1}(m)= \lfloor \tfrac{m}{1+a} + \tfrac12 \rfloor. \end{equation}
I was wondering if there is a quick way to check this.