Suppose $f$ is a function of $\mathbb{R}$ and satisfies $$\forall \ x_0 \in \mathbb{R},\ \exists \ \delta > 0,\ f(x_0) \geq f(x),\ \forall x \in (x_0 - \delta, x_0 + \delta).$$
Prove that there exists a nondegenerate interval $I$, $f$ is constant over $I$.
There are no conditions on the continuity of $f$, so I don't know where to start. I tried the principle of nested intervals but failed.
Any kind of help is appreciated and thank you very much in advance!
From the answer linked in comments, we know $f$ attains at most countably many values. Thus by formulation BCT3 of the Baire category theorem, we know there exists some $y\in \mathbb{R}$ such that $f^{-1}(y)$ is not nowhere dense. That is there exists a closed interval $I$, such that $f^{-1}(y)$ is dense on $I$.
Thus the minimum value of $f$ on $I$ is $y$, as no smaller value can be a local maximum. Pick $x\in {\rm int}(I)$ with $f(x)=y$. We have some closed interval $x\in J\subseteq I$ with the maximum value of $f$ on $J$ being $y$.
We already know the minimum value of $f$ on $J$ is $y$. Thus $f$ is constant on $J$.
