Qualities of a discontinuous function that can be written as a finite/infinite sum of continuous functions

Question

My question is given a discontinuous function, can it be written as an infinite or finite sum if continuous functions. Here, “continuity” is of course relative to a point in the domain. Also, do the class of discontinuous functions that can be written this way have a common property ?The reason such a concern is being raised by me is in context to Quantum mechanics wherein the wavefunction of a system is written as a finite/infinite sum of continuous basis eigen functions, but I’m worried that it may not be always continuous which is important for us to deal with it’s dynamics.

Answer 1

You can't have uniform convergence but for QM what matters is convergence in $L^2$ norm (Fourier series example).

You can expand an $L^2$ wavefunction in eigenfunctions of the Hamiltonian, propagate each term separately, and sum them to get the correct time evolution. The $L^2$ norm remains constant.