Assume $f(x)$ is a smooth function on $\mathbb{R}$ and $f$ does not vanish on any interval. In other words, $f$ can have zero points but we cannot find any interval $(a, b)$ such that $f(x)=0$ for all $x \in (a, b)$. Denote by $\mathcal{Z} = \{x \in \mathbb{R}, f(x) = 0\}$ the zero point set of $f$. Assume $\mathcal{Z}$ is non-empty.

**Question:** Is it possible that every point in $\mathcal{Z}$ is an accumulation point of $\mathcal{Z}$?

Here $x$ is an accumulation point of $\mathcal{Z}$ means there exists a sequence $\{x_n\}$ such that $x_n \in \mathcal{Z}$, $x_n \neq x$ and $\lim_{n\to \infty}x_n = x$ under the usual Euclidean topology.

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