>I think another way to ask this is, can the amplitude
>of a wave function ever go to zero for all values of it's dependent
Have you read Robin Hanson's 'Mangled Worlds' theory?
According to Hanson's theory:

"The mangled worlds approach to quantum mechanics is a variation on many worlds that tries to resolve the Born rule problem by resorting only to familiar decision theory, familiar physical processes, and a finite number of worlds. The basic idea is that while we have identified physical "decoherence" processes that seem to describe measurements, since they produce decoupled wave components corresponding to different measurement outcomes, these components are in fact not exactly decoupled. And while the deviations from exact decoherence might be very small, the relative size of worlds can be even smaller.

As a result, inexact decoherence can allow large worlds to drive the evolution of very small worlds, "mangling" those worlds. Observers in mangled worlds may fail to exist, or may remember events from larger worlds. In either case, the only outcome frequencies that would be observed would be those from unmangled worlds. Thus worlds below a certain size cutoff would be mangled, and not count when calculating probabilities as the fraction of worlds that see an outcome."

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