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 $\left({a \,.\,.\, b}\right)$.
If $\xi \in \left({a \,.\,.\, b}\right)$, then:
- $f \left({\xi^-}\right)$ and $f \left({\xi^+}\right)$ both exist
and:
- $f \left({x}\right) \ge f \left({\xi^-}\right) \ge f \left({\xi}\right) \ge f \left({\xi^+}\right) \ge f \left({y}\right)$
provided that $a < x < \xi < y < b$.
Proof
$f$ is bounded below on $\left({a \,.\,.\, \xi}\right)$ by $f \left({\xi}\right)$.
By Limit of Decreasing Function, the infimum is $f \left({\xi^-}\right)$.
So it follows that:
- $\forall x \in \left({a \,.\,.\, \xi}\right): f \left({x}\right) \ge f \left({\xi^-}\right) \ge f \left({\xi}\right)$
A similar argument for $\left({\xi \,.\,.\, b}\right)$ holds for the other inequalities.
$\blacksquare$