  There are two basic ways of weakening the definition of the well-known metric
regularity property by fixing one of the points involved in the definition. The
first resulting property is called metric subregularity and has attracted a lot
of attention during the last decades. On the other hand, the latter property
which we call semiregularity can be found under several names and the
corresponding results are scattered in the literature. We provide a
self-contained material gathering and extending the existing theory on the
topic. We demonstrate a clear relationship with other regularity properties,
for example, the equivalence with the so-called openness with a linear rate at
the reference point is shown. In particular cases, we derive necessary and/or
sufficient conditions of both primal and dual type. As an application we study
an inexact Newton-type scheme for generalized equations with not necessarily
differentiable single-valued part.
