Let $\left \{ a_{n}\right \}_{n=1}^{\infty }$ be a strictly increasing bounded sequence of real numbers such that $\lim_{n\rightarrow \infty }a_{n}=A$. Let $f:\left [ a_{1},A \right ]\rightarrow \mathbb{R}$ be a continuous function such that for each positive integer $i$, $f|_{\left [ a_{i},a_{i}+1 \right ]}:\left [ a_{i},a_{i+1} \right ]\rightarrow \mathbb{R}$ is either strictly increasing or strictly decreasing.Consider the set
$B=\left \{ \right.M\in\mathbb{R}|$ there exist infinity many $x\in\left[a_{1},A\right]$ such that $f\left ( x \right )=M\left \} \right.$.
Then the cardinality of $B$ is:
- necessarily $0$
- at most $1$
- possibly greater than $1$, but finite
- possibly infinite
