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How do we know that time and distance are not discrete?

I know that it is believed that energy is discrete, in that it travels in quanta. I was wondering if there is any evidence which either proves or disproves something similar with both time and ...
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Is there a finite unit of distance that we cannot divide past?

If distance could be divided into an infinite no of units or points, then it seems to me that motion would be impossible since a meter for instance, having an infinite no of points within it (and the ...