A transcript of a recent speech by Barack Obama contains the following sentence:
Boston police, firefighters, and first responders as well as the National Guard responded heroically, and continue to do so as we speak.
The usage of continue in this case contrasts what I have learned about the verb to stop, as in
We stopped to let the pedestrians pass the crosswalk, so the car did not move.
We stopped letting the pedestrians pass the crosswalk, so I accelerated.
I suppose that these three examples are grammatically correct. People like me, who learned English as a second language, could misunderstand Obama's usage of continue as
[...] and continue (with doing something) in order to do so
I do not believe this is the actual intention. I conclude that this to do vs. doing issue is not a general pattern, but rather a restricted phenomenon. So my questions are:
Is it correct that the aforementioned phenomenon is only relevant for a few verbs? If yes, could somebody provide a list of these verbs?
Is it helpful, to regard the verb to stop as a homonym, where one version of to stop refers to a process of ceasing movement, while the other version is an auxiliary verb followed by a gerund?