$\begin{align}
& y=x+\sin ax\Rightarrow {y}'=1+a\cos ax \\
& \left| \cos ax \right|\le 1\Rightarrow \left| a\cos ax \right|\le \left| a \right|\Rightarrow -\left| a \right|\le a\cos ax\le \left| a \right| \\
& \Rightarrow 1-\left| a \right|\le 1+a\cos ax\le 1+\left| a \right| \\
& \Rightarrow 1-\left| a \right|\le {y}'\le 1+\left| a \right|\Rightarrow 1-\left| a \right|\ge 0\Rightarrow \left| a \right|\le 1 \\
& \Rightarrow -1\le a\le 1\xrightarrow{a\in Z}a=-1,0,1 \\
\end{align}$