TITLE: Uniqueness of the Laplace Transformation
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: 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.