Richard Feynman in this 60's public lecture claimed it's easy to prove "physical space cannot be discrete automata", otherwise it will soon violate existing physical observations towards the end of this lecture. But he didn't mention any proof or explanation there.
I thought about this further and reasoned under classical analysis framework, only continuous space and time make velocity (not the other feature - position) possible. If space or time is really discrete ontologically, then like Zeno's logic in his famous Zeno's Paradox, an arrow can never move without violating metaphysical "Principle of Continuity"! The essence of Zeno paradox's resolution lies in time and space are continuous, thus you can have possible velocity notion via its position's change along with "measuring" its corresponding time interval, and Zeno tacitly avoided the required velocity notion in any motion to arrive at his famous paradox. If time is discrete automata, then you can only have position along with "counting" its time instants, via counting and summing "countably while even infinitely many" instants with freezing positions, the final position is thus still frozen as logically correctly claimed by Zeno. It's hard to imagine a way here to derive a velocity-like concept without some (infinitesimal) interval notion as later formally introduced as differential with its integral in calculus and coined by Leibniz as dx/dt.
Does Zeno's Paradox hold the simplest key to understand and thus explain the widely-held continuous spacetime belief in physics?