It's not literally continuous, but it is real-time rather than turn-based, and positions of units are essentially floats rather than (a small range of) ints. That makes it effectively continuous (too large to just generate a tree of all possible actions and then prune).
Are you sure positions of units are essentially floats? Given how the units seem to arrange themselves (from what I see), I would guess that it's not close to the full range of floats, and instead there are just a few fractional pixel locations that units snap to. This is just a guess however.
-- If this is the case though, the space could be represented by taking larger integer values (say, a magnitude of 1 or 2 higher) to represent positions at a fractional pixel level (say, in 100ths of a pixel).
Buildings snap to a grid. Units take up space according to a hitbox. Hitbox size varies according to each type of unit (E.g. Thors are huge). This becomes important when dealing with AoE.
Consider a group of mutalisks. If you select-all and issue an attack-command or move-command, the mutalisks will bunch up tight and then disperse. Cf a video on the "magic-box technique".
So I wouldn't be surprised if position-values were floats.
I haven't looked at SC2 specifically, but would be surprised if they were floats; I sometimes read game dev blogs and RTSs on average tend to be implemented using int/fixed-point based positioning, to eliminate floating point imprecision as a source of multiplayer sync issues.
