word usage - "stop to do something" vs. "continue to do something"

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:

1. 
   

   Is it correct that the aforementioned phenomenon is only relevant for a few verbs? If yes, could somebody provide a list of these verbs?

   
   


2. 
   

   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?

   
   

Answer

There are a lot of different kinds of infinitive. The infinitive to smoke in

• He stopped to smoke.

(as Colin and Fluffy have already pointed out) is a Purpose infinitive, a kind of adverbial clause answering Why?, which one may introduce with in order to, or move to the front of the sentence,

• He stopped in order to smoke.


• In order to smoke, he stopped.

in order to distinguish it from more common types of postverbal infinitive, like the to smoke in

• He began/started to smoke.

which is an infinitive object complement clause, a kind of noun clause that is the direct object of began or start, representing the activity (or when generic, the habitual activity) that the subject began or started.

Begin and start are also alike in that they can take infinitive complements, like the sentence above, as well as gerund complements, like the one below

• He began/started smoking.

which is synonymous with the second sentence.

However, stop, unlike begin and start, does not allow an infinitive complement, though it does allow a gerund. Thus, any infinitive following stop can't be an infinitive complement clause, and the next likely reading is as a purpose infinitive clause.

Both are common uses for stop, and this little curlicue helps distinguish them. English, and every language, like anything alive, is full of baroque details like this.