If $f$ is monotone prove it is continuous [duplicate]

Assume $f: [a,b] \to \mathbb{R}$ is a monotone function that satisfies the Intermediate Value Theorem. Prove that $f$ is continuous.

It is sort of confusing how they say it satisfies IVT. Don't only continuous functions satisfy IVT? If instead they mean the condition that if it takes any value between $[a,b]$ then it also take any value between $f(a)$ and $f(b)$, then we will need to use the definition of the monotonicity. Suppose that $x<y$ and $f(x) < f(y)$ for any $x,y \in [a,b]$. How can we use this with IVT here to prove that $f$ is continuous?

Solution 1:

It's not restrictive to assume that $f$ is increasing (otherwise use $-f$). Prove that, for $c\in(a,b)$, $$ \lim_{x\to c^-}f(x)=\sup\{f(x):a\le x<c\}, \qquad \lim_{x\to c^+}f(x)=\inf\{f(x):c<x\le b\}, $$ and that $$ \lim_{x\to a^+}f(x)=\inf\{f(x):a<x\le b\}, \qquad \lim_{x\to b^-}f(x)=\sup\{f(x):a\le x< b\}. $$ Suppose that for some $c\in(a,b)$ the limits from the left and from the right are different and apply the hypothesis about the IVT. Finish up with the values at the extremes $a$ and $b$.