A question of style

“But this detective was not Sherlock Holmes but a real life investigator”. My english teacher says this is a grammatically correct sentence. Still, it seems to me that having already used BUT at the beginning … Dido

This is more a question of style, rather than grammar. Starting a sentence with “but” and then repeating it isn’t great English, but it isn’t technically a grammatical error. Writing “English” with a small “e” is, though!