  We consider a locally compact Hausdorff groupoid $G$ which is graded over a
discrete group. Then the fibre over the identity is an open and closed
subgroupoid $G_e$. We show that both the full and reduced C*-algebras of this
subgroupoid embed isometrically into the full and reduced C*-algebras of $G$;
this extends a theorem of Kaliszewski--Quigg--Raeburn from the \'etale to the
non-\'etale setting. As an application we show that the full and reduced
C*-algebras of $G$ are topologically graded in the sense of Exel, and we
discuss the full and reduced C*-algebras of the associated bundles.
