Injectivity of Laplace Transform/Corollary

Corollary to Injectivity of Laplace Transform
Let $f$, $g$ be functions from $\R_{\ge 0} \to K$ of a real variable $t$, where $K \in \set {\R, \C}$.

Let $f$ and $g$ both admit Laplace transforms.

Suppose that the Laplace transforms $\laptrans f$ and $\laptrans g$ satisfy:


 * $\forall t \ge 0: \laptrans {\map f t} = \laptrans {\map g t}$

Let $f$ and $g$ be continuous everywhere on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\R_{\ge 0}$.

Then $f = g$ everywhere on $\R_{\ge 0}$, except possibly where $f$ or $g$ have discontinuities of the first kind.