Summary: We establish the continuity of the Hardy-Littlewood maximal operator on \(W^{1,p}(\varOmega\)), where \(\varOmega \subset \mathbb R^{n}\) is an arbitrary subdomain and \(1 < p < \infty \). Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces \(F^{p}_{s,q} (\varOmega )\) for \(1 < p,q < \infty \) and \(0 < s < 1\).