It's made very clear in both the book and the movie that there is a long time lag for radio communications with Discovery. For example, in the birthday scene in the movie, the parents' transmission is one way and refers to Frank Poole's previous transmission. This all makes sense, because Clarke was writing hard SF with as much scientific accuracy as possible. Clarke himself was one of the originators of the idea of a communications satellite, so it's not as though he'd be ignorant of things like the fact that radio signals propagate at c.
However, later in the movie, after Dave Bowman passes through the star gate, it's equally clear that he is zooming around faster than the speed of light. The implication is that this is something like a wormhole, which is perfectly consistent with relativity. (The speed limit c is a local one.) IIRC the book is vague and hand-wavy about this sort of thing, but mentions something like an idea that the 1:4:9 ratio of the spatial dimensions of the monolith is actually an infinite sequence extending into infinitely many dimensions. Basically this is an example of Clarke's laws.