There's an existence/uniqueness theorem which states: if a function $f$ is of exponential order, then the Laplace transform of $f$ exists and is unique.
Since there's no theorem in my book which states that the converse implication is true, I've been searching for a non exponential order function which is Laplace transformable. But I can't find any. So here's my question:
Is the converse implication true or false? If it's false, could you show me an example? I really can't find any.
Consider $f(t) = \sum_{j=0}^\infty X_j(t)$ where $X_j(t) = j!$ for $j < t < j + 1/(j!)^2$ and $0$ otherwise. This is not exponentially bounded: $j! \exp(-sj) \to \infty$ as $j \to \infty$ for every real $s$. But for $s > 0$, ${\mathscr L}(f)(s) \le \sum_{j=0}^\infty \exp(-sj)/j! = \exp(\exp(-s))$ converges.