Assume the derivative of a function $f$ does not exist everywhere, let's say that it exists everywhere except on a countable set, and that it is continuous between each two successive points of this contountable set. Assume also that the laplace transform of $f$ exists.
Can we still define the Laplace transform of the "almost everywhere" derivative of $f$ ? And in this situation, can we define a Laplace transform for the right/left derivative of $f$ ? If yes, what is the expression of such transform ?
Let $t_n$, $1\le n\le N$ be the points of discontinuity of $f'$. Then, we have for $t_1\ne 0$
$$\begin{align} \mathscr{L}\{f'\}(s)&=\int_0^{t_1}f'(t)e^{-st}\,dt+\sum_{n=1}^{N-1} \int_{t_n}^{t_{n+1}}f'(t)e^{-st}\,dt+\int_{t_{N}}^\infty f'(t)e^{-st}\,dt\\\\ &=s\int_0^\infty f(t) e^{-st}\,dt-f(0^+)-\sum_{n=1}^{N}(f(t_n^+)-f(t_n^-)) e^{-st_n} \end{align}$$