Timed systems can be considered with two types of semantics – dense time semantics and discrete time semantics. The most typical examples of both of them are real semantics and sampled semantics (i.e., discrete semantics with a fixed time step e). We investigate the relations between real semantics and sampled semantics with respect to different behavioral equivalences. Also, we study decidability of reachability problem for stopwatch automata with sampled semantics. Finally, our main technical contribution is decidability of non-emptiness of a timed automaton ω-language in some sampled semantics (this problem was previously wrongly classified as undecidable). For the proof we employ a novel characterization of reachability relations between configurations of a timed automaton.