“ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ”

Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!

Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies