Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be a real analytic function with infinitely many zeros. Let $a\neq 0$ be a real number. Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$.

Answer 1

Hint: If an analytic function's zero set has a limit point, the function is constantly zero. Use that fact to show that $f^{-1}[\{a\}]$ is a discrete set.