Consider, on the space of marked groups, the map $\mathrm{Res}_{\mathcal{C}}$ which associates to a marked group its greatest residually-$\mathcal{C}$ quotient, for different sets $\mathcal{C}$ of groups. Except for trivial cases, this map is discontinuous. We use the Weihrauch lattice to quantify how discontinuous it is. We show that equational noetherianity of $\mathcal{C}$ and whether the set of residually-$\mathcal{C}$ groups is a quasivariety both can be characterized in terms of the position of $\mathrm{Res}_{\mathcal{C}}$ within the Weihrauch lattice. We give exact classifications of $\mathrm{Res}_{\mathcal{C}}$, for $\mathcal{C}$ one of: the set of finite groups, of nilpotent groups, of $k$-nilpotent groups, $k\ge1$, of finitely presentable groups, of LEF groups, of torsion free groups.