In this sentence "That girl keeps smiling at him!" "Smiling at him" is a participle phrase, which can only function as an adjective, so which part of the sentence it is modifying, it must be the subject girl, which is a noun. As the participial phrase is modifying or complementing the subject can we say here keep function as linking verb?
Or keep (on) doing sth. is a phrasal verb function as main verb, so keep smiling is verb, and at him is adverbial preposition phrase. This one is more plausible, right?