Question 107
Let us consider the functions $f, g : \mathbb{R} \to \mathbb{R}$, where $f$ is continuous. Assume that, for all real numbers $a \lt b \lt c$, there is a sequence $(x_n)_{n \ge 1}$ which converges to $b$ such that the limit $\lim_{n \to \infty} g(x_n)$ exists and we have
$$
f(a) \lt \lim_{n \to \infty} g(x_n) \lt f(c).
$$
a) Give an example of such functions, for which $g$ is discontinuous at any real point.
b) Prove that, if $g$ is a monotone function, then $f = g$.
$$
f(a) \lt \lim_{n \to \infty} g(x_n) \lt f(c).
$$
a) Give an example of such functions, for which $g$ is discontinuous at any real point.
b) Prove that, if $g$ is a monotone function, then $f = g$.