Limit of Monotone Real Function
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Theorem
Increasing Function
Let $f$ be a real function which is increasing and bounded above on the open interval $\openint a b$.
Let the supremum of $f$ on $\openint a b$ be $L$.
Then:
- $\ds \lim_{x \mathop \to b^-} \map f x = L$
where $\ds \lim_{x \mathop \to b^-} \map f x$ is the limit of $f$ from the left at $b$.
Decreasing Function
Let $f$ be a real function which is decreasing and bounded below on the open interval $\openint a b$.
Let the infimum of $f$ on $\openint a b$ be $l$.
Then:
- $\ds \lim_{x \mathop \to b^-} \map f x = l$
where $\ds \lim_{x \mathop \to b^-} \map f x$ is the limit of $f$ from the left at $b$.