From my understanding, the Planck length is the smallest length there can be, and every other length has to be an exact integral multiple of the Planck length [1]. I believe the same applies to the other Planck units [2]- they are fundamental units. Similarly, every charge has to be an exact integral multiple of the charge of the electron. These fundamental units seem to to indicate that our universe is not continuous, but quantized/discrete. With that said, there are some theories stating that these fundamental units only apply to our universe (and even in our own universe, the fundamental units may change over long periods of time), and that other universes may have different fundamental units. So depending on how you define the universe, the answer to whether the universe is continuous or discrete could change.
[1] http://en.wikipedia.org/wiki/Planck_length [2] http://en.wikipedia.org/wiki/Planck_units