In this paper we extend the familiar concept of spatial instability and growth of disturbances in the downstream direction to include spatial instability and growth in the wall-normal direction. The stability theory of boundary layers has generally been concerned with determining the evolution of disturbances inside a boundary layer (this is where disturbances have their largest amplitudes and can cause a laminar boundary layer to become turbulent). Outside a boundary layer, where the basic flow is uniform, normal-mode disturbances decay exponentially with distance from the wall to satisfy homogeneous boundary conditions. In this paper we present a surprising scenario where an impulsive disturbance, made up of a superposition of these normal modes, nonetheless grows exponentially with distance from the wall. While the usual convective instability with exponential growth in the downstream direction can be efficiently characterized by spatial modes with complex wavenumbers, the new convective instability can be efficiently characterized by modes with exponentially diverging ‘eigenfunctions’ obtained by moving certain branch-cuts in the complex wavenumber plane. The new instability is therefore associated with an interaction between the discrete spectrum and the continuous spectrum. We emphasize, however, that the homogeneous boundary conditions are always satisfied, and that at any finite time exponential growth only occurs over a finite distance from the wall, but this distance increases linearly with time. Interactions between poles and branch-cuts have been found before, but the results presented here provide a physical interpretation for this spectral behaviour. A further curiosity is that some of these divergent modes have been found to violate Howard's semi-circle theorem.