Is the universe discrete or continuous?

What proof is there that the universe is discrete and what proof is there that the universe is continuous?

You can never know if the universe is discrete for certain since even if it appears so there may still be some undetectable phenomena you are unaware of.

You can never find evidence to suggest the universe is continuous as there can never be a scientific instrument capable of infinite accuracy.

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In the interest of being open minded, couldn't the same things conceivably have been said of relativity during its conception?  Not suggesting that string theory is or isn't true here; just remarking.  If string theory produced a model that seemed to fit with what all we've seen, and predicted something else we hadn't previously known and could test, then it would be useful.  What all of physics lately hasn't been useful anyway?  Nuclear stuff and quantum physics are fairly recent and have produced huge effect on daily life.