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 on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\left [{0 \,.\,.\, \to} \right)$.

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:

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

Then $f = g$ everywhere on their domains, except possibly where $f$ or $g$ have discontinuities of the first kind.

Corollary
Suppose $f$ and $g$ are continuous everywhere on their domain.

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