Q:

Understanding a theorem regarding to monotonic functions

Let $f$ be a monotonic on an interval $I$. if $x_0$ is interior to $I$, then the one-sided limits $\lim_{x\to {x_0}^-}f(x)$ and $\lim_{x\to {x_0}^+}f(x)$, both exists.

Suppose the theorem is true, and interval $I=(a,b)$. Is it implying that the function has one-sided limits at the edges? (which are $a$ and $b$).

A:

No, it is not implying that the function has one-sided limits at the edges. It specifies that $x_0$ is interior to the interval $I$. This means that $x_0$ cannot exist at either point $a$ or $b$.

The interior of an interval I is the largest open interval that is contained in I; it is also the set of points in I which are not endpoints of I. 

For more information on the terminology, see the Wikipedia page on the topic.

