I would do several things:

1. Make a printout of the flight status immediately (i.e. before it is no longer available). It shows that the plane departed early at 6:02pm instead of the scheduled time of 6:10pm.

2. Check to see that Continental does not have a rule requiring check in at least 30 minutes prior to the flight. This is (or at least used to be) a typical rule that most US domestic airlines had.

If there is no 30 minute check in rule, I would contact Continental Customer Service and ask for denied boarding compensation, especially since the flight departed early. It is still possible to make it from the check in desk to the departure gate in that amount of time if the security line is not too long (although at this time on a Thursday evening it might well be too long).