  We investigate Sarnak's M\"obius Disjointness Conjecture through
asymptotically periodic functions. It is shown that Sarnak's conjecture for
rigid dynamical systems is equivalent to the disjointness of M\"obius from
asymptotically periodic functions. We give sufficient conditions and a partial
answer to the later one. As an application, we show that Sarnak's conjecture
holds for a class of rigid dynamical systems, which improves an earlier result
of Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.
