necessary condition of Laplace transform

Question

from the book of engineering mathematics by G.Zill, I read That

"The inverse Laplace transform of a function F(s) may not be unique; in other words, it is possible that Laplace transform of f1 and f2 is same and yet f1 and f2 is different. For our purposes this is not any- thing to be concerned about."

I found if f1 and f2 is different at countable many points, he is right because of Riemann integral's property.

1. however, I still wonder if there are any other case. if there are, how can we know that? (And I think this question is related to the question of the necessary condition of Lapace transform.)
2. And if so, why does the inverse Laplace transform have Linearity even though their output is differ case by case?

Answer 1

Given that the (one-sided) Laplace transform is defined by $$ F(s) := \int_0^\infty f(t)\,e^{-st} \,\mathrm{d}t, $$ it doesn't take into account the points on the negative part of the real axis, since they lie outside the domain of integration. In consequence, functions can differ on $(-\infty,0)$ and have the same Laplace transform at the same time.

A simple example is the constant function $f(t) = 1$ in comparison with the Heaviside function $\theta(t) = 1_{(0,\infty)}(t)$, which differ at an uncoutably infinite number of points while having the same transform (namely $1/s$).

As for the linearity of the Laplace transform (and its inverse), it stems simply from the linearity of the integral towards its integrand.