Anon, We can approach this type of structure this way : I saw the boy- is a complete sentence * ( reading a book). * The whole phrase in parenthesis is considered an adverbial clause consisting a present participle which modifies the noun, "boy". There are many examples in our everyday life.
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Anonymous... describe the grammatical function of the word "reading" in the following sentence:I'd say it's the verb of the clause "the boy reading".
I saw the boy reading.