I don't know if this was mentioned before, but here we go. In the first firm, Daniel asks Miyagi if he ever taught anyone karate before and Miyagi said no. Then in this film, he tells Julie "Miyagi teach your grandfather, your grandfather teach your father, your father teach you." Any thoughts?
as stupid as it might sound, this film aint a continuation from the 3 previous karate kids, this is just a whole neew different 1 all togetha, dats y its called the next karate kid, not the karate kid part 4 or sumfin. (hope dat helped but i dunno if its true, it was just a thought)
I believe that Miyagi said "No" to Daniel in the first film because if he had said "Yes", it would have encouraged Daniel to ask him to ask for lessons. Plus, as Miyagi-sensei went on to tell Daniel in "Karate Kid II", his father had made an exception to the family rule about Miyagi karate only being taught within the family in order to teach Sato. Now it was certainly Miyagi's choice as to who to teach, once the tradition had been handed down to him, particularly since he had no children of his own, and in teaching Julie's grandfather during WWII, that was a choice he was entitled to make.
And yes, it's easy to say "Oh yeah, they made a continuity gaffe", but I really do not believe Miyagi would have said "Yes" to Daniel's question.