Working on an assignment today, I thought of a problem I haven't been able to solve and haven't been able to find any solutions for online.
What would the Laplace Transform of a rounded function be? So for instance, let's say we wanted to take the LT of $sin(x)$ rounded to the nearest 0.1. I know there are ways of calculating the LT of discontinuous functions using the unit step function, but none of the methods I've found seem scalable to functions with many, or even infinite, discontinuities.
Actually, this looks somewhat messy. You can represent the proposed rounding function as
$$f(x) = \frac1{10} \left \lfloor 10 \sin{x}+\frac12 \right\rfloor $$
Using the fact that
$$\lfloor y \rfloor = y - \frac12 +\frac1{\pi} \sum_{k=1}^{\infty} \frac{\sin{2 \pi k y}}{k}$$
we get, as our LT:
$$F(p) = \int_0^{\infty} dx \, \sin{x} \, e^{-p x} + \frac1{10 \pi} \sum_{k=1}^{\infty} \frac{(-1)^k}{k} \int_0^{\infty} dx \,\sin{(20 \pi k \sin{x})} \, e^{-p x}$$
The integral on the left is the usual LT for the sine. The integral on the right, the correction for the rounding, looks like a hell of Bessel functions. If I find any better representation, I will post it.