Laplace Transform Exists if Function Piecewise Continuous and of Exponential Order

Theorem
Let $f$ be a real function which is:
 * piecewise continuous in every closed interval $\closedint 0 N$
 * of exponential order $\gamma$ for $t > N$

Then the Laplace transform $\map F s$ of $\map f t$ exists for all $s > \gamma$.

Also see

 * Laplace Transform of Reciprocal Square Root for a real function which is not of exponential order but which does have a Laplace transform