Existence of inverse Laplace transform of $F(1/s)$

Question

I have two questions about inverse Laplace transform.

Given a function $F(s)$, does its inverse Laplace transform always exists?

If it's not, assume $F(s)$ has an inverse Laplace transform, does the inverse Laplace transform of $Q(s) = F(1/s)$ exists?

Answer 1

You have to have F(s)-->0 as s-->Inf because of how the Laplace transform is defined (integrate f(t) with exp(-st) over all positive t's). As s-->Inf, the exponential factor goes to zero as s --> Inf for all positive t's, so you have to have for any Laplace-transformable f(t) the corresponding transform F(s) --> 0 as s-->Inf. This is a necessary condition, but I'm not sure if sufficient.

This precludes the existence of inverse Laplace transforms of functions such as F(s) = s, sqrt(s) etc.