Limit of Monotone Real Function/Decreasing/Corollary
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Corollary to Limit of Decreasing Function
Let $f$ be a real function which is decreasing on the open interval $\openint a b$.
If $\xi \in \openint a b$, then:
- $\map f {\xi^-}$ and $\map f {\xi^+}$ both exist
and:
- $\map f x \ge \map f {\xi^-} \ge \map f \xi \ge \map f {\xi^+} \ge \map f y$
provided that $a < x < \xi < y < b$.
Proof
$f$ is bounded below on $\openint a \xi$ by $\map f \xi$.
By Limit of Decreasing Function, the infimum is $\map f {\xi^-}$.
So it follows that:
- $\forall x \in \openint a \xi: \map f x \ge \map f {\xi^-} \ge \map f \xi$
A similar argument for $\openint \xi b$ holds for the other inequalities.
$\blacksquare$