Question 25
Let $f:\mathbb R\to\mathbb R$ be monotone.
a) Prove that $f$ has one-sided limits at every $x_0\in\mathbb R$.
b) Define $g:\mathbb R\to\mathbb R$ by $g(x)=\lim_{t\gt x}f(t)$, i.e. $g(x)$ is the left-hand limit at $x$. Prove that if $g$ is continuous, then $f$ is continuous.
a) Prove that $f$ has one-sided limits at every $x_0\in\mathbb R$.
b) Define $g:\mathbb R\to\mathbb R$ by $g(x)=\lim_{t\gt x}f(t)$, i.e. $g(x)$ is the left-hand limit at $x$. Prove that if $g$ is continuous, then $f$ is continuous.