Suppose that a real-valued function function $f$ defined on an open interval $I$ has the property that at every point $a \in I$ that either the limit of $f$ as $x$ approaches $a$ from the left exists or the limit of $f$ as $x$ approaches $a$ from the right exists (or both). In this context, I am trying to understand why the following statement must be true:
For every $\epsilon > 0$ there exists either an open interval $J_L = (L, a) \subset I$ such that
$ |f(s) - f(t)| < \epsilon \; \; \forall s,t \in J_L $
or an open interval $J_R = (a, R) \subset I$ such that $ |f(s) - f(t)| < \epsilon \; \; \forall s,t \in J_R $
How can I see that this is a true statement?