Function that satisfies intermediate value property and $f(x) ≤ f(y)$ for all $x<y$ is continuous

Question

A function $f$ is increasing on $A$ if $f(x) ≤ f(y)$ for all $x<y$ in $A$. Show that if $f$ is increasing on $[a, b]$ and satisfies the intermediate value property, then $f$ is continuous on $[a, b]$.

My attempt: Supose it is not continuous. Take $|x-c| < \sigma$ but $|f(x)-f(c)| \ge \epsilon_0$ than $|f(x)| \ge \epsilon_0 + |f(c)|$, so $f(x) >f(c)$ which is false.

Answer 1

Hint: show that, for every $c\in(a,b)$, both $$ \lim_{x\to c^-}f(x)\qquad\text{and}\qquad\lim_{x\to c^+}f(x) $$ exist finite. For instance, $$ \lim_{x\to c^-}f(x)=\sup_{a\le x<c}f(x) $$ What if the two limits differ?

Finish up with the special cases of $c=a$ and $c=b$.