Exercise 4.6.5

Let $f$ be monotone and assume $f$ is increasing. For some $c$ we want to show $\underset{x\to c^{\text{−}}}{\lim <!--\; FUNCTION\; APPLICATION\; -->}f(x)$ and $\underset{x\to c^{\text{+}}}{\lim <!--\; FUNCTION\; APPLICATION\; -->}f(x)$ exist.

Let $𝜖>0$ and set $L=\sup <!--\; FUNCTION\; APPLICATION\; -->\{f(x):x<c\}$, by the definition of $\sup <!--\; FUNCTION\; APPLICATION\; -->$, $L-𝜖$ is not an upper bound for $\{f(x):x<c\}$, hence there exists a ${\delta }_1>0$ with $f(c-{\delta }_1)>L-𝜖$, thus $0<c-x<{\delta }_1$ implies $|f(x)-L|<𝜖$ (this is where we use the fact that $f$ is increasing!), hence the lower limit exists. Likewise for $M=\inf <!--\; FUNCTION\; APPLICATION\; -->\{f(x):x>c\}$ we get ${\delta }_2>0$ with $0<x-c<{\delta }_2$ implying $|f(x)-M|<𝜖$, hence the upper limit exists.

Putting these together, we see that $f$ is continuous at $c$ if and only if $L=M$. In other words, the only possible discontinuity is a jump discontinuity $L\ne M$.