In an application of the Implicit Function theorem, I asked myself the following question.
Let $f\colon [a,b] \times \mathbb R \times \mathbb R \to \mathbb R$ be a continuously differentiable function on $(a,b) \times \mathbb R \times \mathbb R$ such that $f$ as well as all partial derivatives $\partial_if$ have continuous extensions to $[a,b] \times \mathbb R \times \mathbb R$. Assume that $f(x,u,\xi) = 0$ and $\partial_\xi f(x,u,\xi) > 0$ for all $(x,u,\xi) \in [a,b] \times \mathbb R \times \mathbb R$. By the implicit function theorem there exists a unique continuously differentiable $g:(a,b) \times \mathbb R \times \mathbb R \to \mathbb R$ such that $f(x,u,g(x,u)) = 0$.
Question: Can I also extend $g$ as well as all partials of $g$ continously to $[a,b] \times \mathbb R \times \mathbb R$? I assume I can at least for the partials since the implicit function theorem also provides me with an explicit formula for the latter from which I might be able to conclude. But what about $g$ itself?