My Laplace transform textbook's definition of improper integral is
The improper integral is defined in the obvious way by taking the limit:
$\lim\limits_{R \to \infty} \int\limits_a^R F(x) \ dx = \int\limits^\infty_a F(x) \ dx$
provided $F(x)$ is continuous in the interval $a \le x \le R$ for every $R$, and the limit on the left exists.
Shouldn't the condition be that $F(x)$ has at most finitely many jump discontinuities (that is, that $F(x)$ is piecewise continuous), as is the requirement for Riemann integrability?
Riemann integrability is a weaker condition than continuity on a closed interval [a, R] so the definition is just a little stronger than required.