  The operational definition of a weak value for a quantum mechanical system
involves the limit of the weak measurement strength tending to zero. I study
how this limit compares to the situation for the undisturbed (no weak
measurement) system. Under certain conditions, which I investigate, this limit
is discontinuous in the sense that it does not merge smoothly to the Hilbert
space description of the undisturbed system. Hence, in these discontinuous
cases, the weak value does not represent the undisturbed system. As a result,
conclusions drawn from such weak values regarding the properties of the studied
system cannot be upheld. Examples are given.
