Showing the Cumulative distribution only has countable plateaus

Question

Let $F$ be a cumulative distribution function. Show that $F$ only has countably many plateaus.

My idea: Define $A_{n}:=\{[a,b]\subseteq \mathbb R: [a,b]$ is a plateau of length $\geq \frac{1}{n}\}$

I want to prove $|A_{n}|<\infty$, so I assume $|A_{n}|=\infty$

This means there exists $([a_{i},b_{i}])_{i\in \mathbb N}$ so that $\lambda([a_{1},b_{1}])\leq...\leq\lambda([a_{k},b_{k}])$

But I do not know whether I am on the right track here, and how to continue

Answer 1

Let $G(q):=\inf\{x\in \mathbb{R}:F(x)\ge q\}$, $0<q<1$ be the corresponding quantile function. A flat region of $F$ corresponds to a jump of $G$. Since $G$ is non-decreasing on $(0,1)$, it can have at most countable number of jumps.

Answer 2

To each plateau you can associate a rational number (namely, some rational number in the interval where your function takes a constant value). These rational numbers are different for each plateau. Therefore, there is an injective mapping from the set of plateaus into the rational numbers. As a consequence, the number of plateaus is at most countable.

Actually, this property does not have anything to do with the non-decreasing nature of the cdf.