Does Lorentz Invariance depend on the continuity of spacetime?

Question

We may also ask: Is the assumption of continuity of spacetime required by Lorentz Invariance?

Answer 1

Yes. The Lorentz group contains continuous (i.e. arbitrarily small) rotations and boosts, and any discretization of spacetime would fail to be invariant under rotations or boosts smaller than the scale of, say, the primitive unit cell (if the discretization forms a lattice).