[NOT A TEACHER AT ALL]
Stop + gerund means to finish an action in progress:
1) I stopped working for them because the wages were so low.
2) Stop tickling me!
Stop + to-infinitive means to interrupt an activity in order to do something else, so the infinitive is used to express a purpose:
1) I stopped to have lunch. (= I was working, or travelling, and I interrupted what I was doing in order to eat.)
2) It's difficult to concentrate on what you are doing if you have to stop to answer the phone every five minutes.
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