Question 22
Let $f:\mathbb R\to\mathbb R$ be monotone.
a) Prove that $f$ has right-hand and left-hand limits at every $x_0\in\mathbb R$.
b) Define $g(x)=\lim_{t\nearrow x}f(t)$, the left-hand limit of $f$ at $x$. Prove that if $g$ is continuous then $f$ is continuous.
a) Prove that $f$ has right-hand and left-hand limits at every $x_0\in\mathbb R$.
b) Define $g(x)=\lim_{t\nearrow x}f(t)$, the left-hand limit of $f$ at $x$. Prove that if $g$ is continuous then $f$ is continuous.