Question 7
Suppose $f: \mathbb{R} \to \mathbb{R}$ is a monotonic function.
a. Prove that $f$ has one-sided limits at any point $x_0 \in \mathbb{R}$.
b. Define the function $g: \mathbb{R} \to \mathbb{R}$, $g(x) = \lim_{t \to x} f(t)$, i.e. $g(x)$ is the left-sided limit at $x$ of the function $f$. Prove that $g$ is a continuous function, then $f$ is also continuous.
a. Prove that $f$ has one-sided limits at any point $x_0 \in \mathbb{R}$.
b. Define the function $g: \mathbb{R} \to \mathbb{R}$, $g(x) = \lim_{t \to x} f(t)$, i.e. $g(x)$ is the left-sided limit at $x$ of the function $f$. Prove that $g$ is a continuous function, then $f$ is also continuous.