What with quantum mechanics and all, discrete energy levels ect…
Are we sure that anything can be classified as continuous?
Is there some fundamental ‘unit’ of time? That doesn’t seem to sit well with me with regards to a load of things.
Can a particle move an infinately small distance?
Maybe - Google “planck time”.
Look at the article on Loop Quantum Gravity is Jan. 2004 Scientific American. It talks of quantized time and space. The “atom” of length is 10^{-33} cm and therefore the “atom” of time is 10^{-43} sec. In geometric units, they are the same (approximately) since time and distance are calculated in the same units, taking the speed of light as a dimensionless 1.
I know I’m a century ahead of my time, but I think all of the current particles will turn out to be made up of others.
And all the quantum jumps will turn out to be smooth glides, just too fast to measure in this century.
My prediction, based mainly on this concept:
A dog has fleas and those fleas bite em,
And those fleas have fleas and they too bite em,
And those fleas too have fleas that bite em,
And so proceed ad infinitum. 
The essence of quantum mechanics is the assumption that it is not possible to always find a more subtle probe of a system. If this assumption is correct, then we will probably never know the structure of the Universe on the smallest length and time scales, about which you are asking. I say “probably” because while it would be by definition impossible to experimentally probe those length and time scales, it is possible that the Universe contains no adjustable parameters: that is to say, the physical nature of the Universe at all length scales may be determinable by pure theory because it is required by logical consistency from what we can observe. Like the old joke, it may be possible to deduce everything about the nature of the Universe merely from the fact that it exists.
Although, I find it unlikely in this case that we are, as a species, intelligent enough to comprehend the whole picture of things, let alone summon the logic to deduce it.
That is, I think it’s as likely that a human being will ever be able to answer this question as that a horse should learn calculus.