Let $V:\mathbb{R}^2\rightarrow\mathbb{R}$ be continuously differentiable and positive definite. I'm currently reading a book where the authors claim that $V$ has a continuum of closed level curves around the origin. While I certainly find this to be plausible it is not immediately evident to me. How is this proven? If the proof is extensive a simple sketch will suffice and I will do my best to fill in the gaps.
Note that in this context $V$ being positive definite means that $V(\textbf{0})=0$ and $\textbf{x}\in\mathbb{R}^2\setminus\{\textbf{0}\}\Rightarrow V(\textbf{x}) > 0$.