We study the reachability problem for continuous one-counter automata, COCA for short. In such automata,
transitions are guarded by upper and lower bound tests against the counter value. Additionally, the counter
updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor
between zero and one. Our three main results are as follows: (1) We prove that the reachability problem for
COCA with global upper and lower bound tests is in NC2; (2) that, in general, the problem is decidable in
polynomial time; and (3) that it is NP-complete for COCA with parametric counter updates and bound tests.