We are wanting to prove a function is smooth and wondering if the implicit function theorem is useful for such things.
More concretely, suppose that $F\colon \mathbb R^2 \to \mathbb R\colon (x,y) \mapsto F(x,y)$ is a smooth function. Let $f\colon \mathbb R \to \mathbb R$ be such that $$F(x,f(x)) = 0$$ and $\partial_y F(x,f(x)) \neq 0$. Does the implicit function theorem automatically imply that $f$ is smooth?