Limit of Monotone Real Function/Decreasing/Corollary

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.

$\blacksquare$