Now at MO.
Consider the structure $(\mathbb{R};+,-,*,0,1,<)$. We adjoin to it a unary function $f$ defined everywhere on the real line, to form the expanded structure $(\mathbb{R};+,-,*,0,1,<,f)$ I can define that $f$ is everywhere continuous by a complicated definition involving several quantifiers. What is the minimum quantifier complexity that would suffice to give an equivalent definition, along with the proof that it is minimal? I apologize if my question is insufficiently rigorous or precise.