Today; when I was doing some Inverse Laplace transformation in the class, I encountered the following problem cited in Zill's book:
The inverse Laplace transformation may be not unique. In Problems 29 and 30 evaluate $\mathscr{L}\{f(t)\}$.
29. $f(t) = \left\{ \begin{array}{ll} 1, & \quad t\ge0,~ t\ne1, t\ne2 \\ 3, & \quad t=1 \\ 4, & \quad t=2 \\ \end{array} \right.$
30. $f(t) = \left\{ \begin{array}{ll} \text{e}^{3t}, & \quad t\ge0,~ t\ne 5, \\ 1, & \quad t=5 \end{array} \right.$
I did them, but I was wondered how to explain the students if they ask me about the criteria for the uniqueness. I think it is rooted in the functional analysis, however, I am weak in this area. Is there any easier way to explain the uniqueness for second year graduate students? Thanks for your time.
The issue here is continuity, since you typically cannot hope to recover the behavior of a function at a point which is not a point of continuity from an integral transform. If you consider continuous functions which decay fast enough (for example, subexponentially decaying functions) then you can prove uniqueness fairly easily by a standard calculation. For example, see here.