Given $\mathbf{F}: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m$ which is continuously differentiable, consider a system of equations such that $\mathbf{F}(\mathbf{x}, \mathbf{y}) = 0$. Suppose that there exists an implicitly defined unique function $\mathbf{y}^*: \mathbb{R}^n \to \mathbb{R}^m$ such that $\mathbf{F}(\mathbf{x}, \mathbf{y}^*(\mathbf{x})) = 0$ (i.e., I do not have a closed form for it but I know that for each $\mathbf{x}$ there exists a unique $\mathbf{y}$ such that $\mathbf{F}(\mathbf{x}, \mathbf{y}) = 0$) , and for all $\mathbf{x} \in \mathbb{R}^n$, the Jacobian $\nabla_\mathbf{y} \mathbf{F}(\mathbf{x}, \mathbf{y}^*(\mathbf{x}))$ is invertible.
By the implicit function theorem, I know that there for each $\mathbf{x} \in \mathbb{R}^n$ exists a set $\mathbf{x} \in U \subset \mathbb{R}^n$ such that the unique restriction of $\mathbf{y}^*$ to $U$ is continuously differentiable.
Does the existence of a globally unique $\mathbf{y}^*$ imply the global continuous differentiability of $\mathbf{y}^*$? That is, can I stitch up the locally continuous derivative maps to obtain a globally continuous derivative map?