Suppose you have a function $h: \mathbb{R}_{+} \to \mathbb{R}_{+}$ that is monotone decreasing (thus bounded) and locally integrable. Can we conclude that it is piecewise continuous?
Obviously, due to the monotonicity there are only countably many discontinuities, but that is not good enough. There could be accumulation points, where the piecewise continuity breaks apart. But maybe there is some workaround.
What I am really interested in, though, is whether the Laplace transform of those functions is unique, i.e. that I can conclude $h_1 = h_2$ for such functions $h_1,h_2$ that have the same Laplace transform in an interval $\lambda \in [0,\Lambda]$.