Injectivity of Laplace Transform/Corollary

Corollary to Injectivity of Laplace Transform
Let $f$, $g$ be functions from $\left [{0 \,.\,.\, \to} \right ) \to \mathbb F$ of a real variable $t$, where $\mathbb F \in \left\{ {\R, \C}\right\}$.

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

Suppose that the Laplace transforms $\mathcal L \left\{{f}\right\}$ and $\mathcal L \left\{{g}\right\}$ satisfy:


 * $\forall t \ge 0: \mathcal L \left\{{f\left({t}\right)}\right\} = \mathcal L \left\{{g\left({t}\right)}\right\}$

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 $\left [{0 \,.\,.\, \to} \right)$.

Then $f = g$ everywhere on $\left [{0 \,.\,.\, \to} \right )$, except possibly where $f$ or $g$ have discontinuities of the first kind.