Injectivity of Laplace Transform

Theorem
Let $f$, $g$ be functions from $\R_{\ge 0} \to K$ of a real variable $t$, where $K \in \set {\R, \C}$.

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

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

Suppose that the Laplace transforms $\laptrans f$ and $\laptrans g$ satisfy:


 * $\forall t \in \R_{\ge 0}: \laptrans {\map f t} = \laptrans {\map g t}$

Then $f = g$ everywhere on $\R_{\ge 0}$.