A simple question.
Suppose that a function $f$ is $L^1_\text{loc}([0,\infty[)$ and of exponential order. Then the Laplace transform of $f$ always exists because its satisfies a notorious sufficient condition for the existence of Laplace Transform. For this type of function, is the following identity always valid? $$f(s)=\mathcal{L}\{ \mathcal{L}^{-1}\{f\}\}=\int^\infty_0 e^{-st} \mathcal{L}^{-1}\{f(s)\}(t) \, dt $$ In other words, if I know that a function $f$ admits Laplace Transform, can I always express $f$ as the Laplace Transform of its Inverse Laplace Transform or I need further conditions?
Thanks