Inverse Laplace transform of functions with jump discontinuities

Question

Given a function $F(s)$, suppose we define its inverse Laplace transform as:

\begin{equation} f(t) = \lim_{k \to \infty} \frac{(-1)^{k}}{k!}\left(\frac{k}{t}\right)^{k+1}F^{(k)}\left( \frac{k}{t} \right) \end{equation}

for $t > 0$.

Now suppose we take the ordinary Laplace transform of a function with a jump discontinuity. In this case, what would the above formula recover when applied to the Laplace transform of the function at the jump discontinuity? Would it be $\frac{1}{2}(f(x^+) + f(x^-))$?