A Note on the Inverse Laplace Transformation of $f(t)$

Let $\mathcal{L}\{f(t)\} = \int_{0}^{\infty}e^{-st}f(t)dt$ denote the Laplace transform of $f$. It is well-known that if $f(t)$ is a piecewise continuous function on the interval $t:[0,\infty)$ and of exponential order for $t > N$; then $\lim_{s\to\infty}F(s) = 0$, where $F(s) = \mathcal{L}\{f(t)\}$. In this paper we prove that the lesser known converse does not hold true; namely, if $F(s)$ is a continuous function in terms of $s$ for which $\lim_{s\to\infty}F(s) = 0$, then it does not follow that $F(s)$ is the Laplace transform of a piecewise continuous function of exponential order.

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