Ill-posedness of real Laplace inversion

Summary

Recovering a function $f$ from the values of its Laplace transform $F(s)$ at real points $s > 0$ is an ill-posed problem in the sense of Hadamard. The solution is unique, but it does not depend continuously on the data. Transforms that are very close can come from functions that are far apart. So small errors in measured values of $F$ can produce large errors in the computed $f$, and numerical inversion needs regularisation. The problem fails the stability condition in the definition of a well-posed problem.