Does doxycycline hyclate 100mg twice daily continue working after completing the 10 day course,and if so, for how long ?

It depends....: Doxycycline is used for killing germs, but also for changing the reparative process in the skin. It's ability to kill bacteria likely wanes very quickly (within a day) after the last dose. However the benefits in the skin may continue for weeks. Intermittent courses of doxycycline can be helpful for acne, for this reason.

MED KINETICS: Unfortunately medication does not last in the bloodstream beyond a certain time based upon the drug's pharmacokinetics. Beyond the period of time you mentioned this drug will typically only last about 22 hours before it is no longer detectable.