Bounded and  Monotonic Functions

Definition: We say that a  $f$  is bounded for a set  D,there is a number $B \gt 0$ where for all $x \in D, |f(x)| < B$
Theorem: CCD.LB:
If $\lim_{x \to a}f(x)= L$ then in some interval $(r,s)$ with $a \in (r,s)$ there is a number $B \gt 0$ where for all $x \in (r,s), |f(x)| < B$.

Proof of Theorem CCD.LB (with mapping diagram).

Theorem: CCD.BML: Suppose $f$ is monotonic and bounded on a set D and $a \in D$. Then there exist $L$ and $M$ where $\lim_{x \to a^+}f(x)= L$ , $\lim_{x \to a^-}f(x)= M$.
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