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