대수경 17-5

Show that function $f:\mathbb{R} \to \mathbb{R}$ has countable local maximum points.

Proof. Define $M$ by a set of local maximum points of $f$. Note that $\mathbb{R} = \cup_{n\in \mathbb{Z}} [n, n+1]$.

We will show that for every bounded interval $[a, b]\subset \mathbb{R}, M\cap [a, b]$ is countable. 

Define $M_{k}=\{x\in M\cap [a, b] : |x-y|<1/k \Rightarrow f(x)>f(y) \}$.

Then $M_k$ is finite for all $k\in \mathbb{N}$. If not, there is a limit point of $M_k$ because $M_k$ is bounded. However, this is impossible because for distinct $x_{1}, x_{2}\in M_{k}, |x_{1}-x_{2}|\geq 1/k$ by definition of $M_{k}$. 

So $M\cap [a, b]=\cup_{k\in \mathbb{N}} M_{k}$ is countable union of finite sets, so is countable.

Consequently, $M=\cup_{n\in \mathbb{Z}} M\cap [n, n+1]$ is countable union of countable sets, so is countable.