Injectivity of Laplace Transform

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

Further let $f$ and $g$ be continuous everywhere on their domains.

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\} $

Then $f = g$ everywhere on $\left [{0 \,.\,.\, \to} \right )$

Corollary
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.