Continuity of Phase Shift at Continuum Bound State

When a separable potential has a bound state in the continuum, there are two possible definitions of the phase shift. It is shown that if the phase shift is chosen to be the boundary value of the phase of an analytic function, then it has a jump discontinuity of magnitude $\pi$ at each continuum bound state. Some advantages of this definition, as opposed to one giving a continuous phase shift, are presented.