Abstract A class of set-valued mappings called linearly semi-open mappings is introduced which properly contains the class of linearly open set-valued mappings. A stability result for linearly semi-open mappings is established. The main result is a Lyusternik type theorem. Sufficient conditions for linear semi-openness of processes are derived. To verify these conditions, the openness bounds of certain processes are computed. A representation of the openness bound of a locally Lipschitz function in a Clarke non-critical point is given. It is shown that continuous piecewise linear functions on R n are linearly semi-open under certain algebraic conditions.