Well, it depends on your definition of “continuous”.
What it means is that spacetime is quantized–that at certain levels, it comes in discreet little packages that can no longer be subdivided. Originally, Planck did it as a math trick when trying to solve the Ultraviolet Catastrophe, but it had the annoying characteristic of being essential.
Is spacetime continuous? Yeah, pretty much. See, at that size, “location” is not really a thing. Quantum particles are everywhere they have a possibility of being all at once, so what does “location” even *mean* in that context?
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