The authors consider the equation \[ Lu:= \text{div}(|\nabla u|^{p(x)- 2}\nabla u)= 0,\tag{1} \] where \(p(x)\) is a function measurable in \(\Omega\) such that \[ 1< p_1\leq p(x)\leq p_2< \infty. \] The authors investigate the behaviour at boundary points of a solution of the Dirichlet problem associated to (1), where \(\Omega\) is a bounded domain and \(p(x)\) satisfies some natural additional assumptions. Moreover, they obtain a regularity criterion for a boundary point of Wiener type, an estimate for the modulus of continuity of the solution near a regular boundary joint, and geometric conditions for regularity.