  In this paper, we introduce the concept of semi-continuous $g$-frames in
Hilbert spaces. We first construct an example of semi-continuous $g$-frames
using the Fourier transform of the Heisenberg group and study the structure of
such frames. Then, as an application we provide some fundamental identities and
inequalities for semi-continuous $g$-frames. Finally, we present a classical
perturbation result and prove that semi-continuous $g$-frames are stable under
small perturbations.
