There is "Sufficient Condition Theorem" to test if a function's Laplace transform exists.
If a function $f$ is piecewise continuous on $[0, \infty)$ and is of exponential order $\alpha$, then the Laplace transform of $f$ exists for $s>\alpha$.
I read a document stating that this sufficient condition may not be necessary, i.e., a function which is not piecewise continuous on $[0,\infty)$ can also have its Laplace transform exist. Then take $f(t)=t^{-\frac12}$ for example.
However, $f(t)$ will have no definition at $t=0$ therefore, I am not completely convinced by the example.
Does it mean that all continous functions have to be defined in a closed domain?
Then what other examples can be more persuasive to the sufficient condition theorem?