Injectivity of Laplace Transform/Corollary

Corollary to Injectivity of Laplace Transform
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.