Limit of Monotone Real Function/Increasing/Corollary

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Corollary to Limit of Increasing Function

Let $f$ be a real function which is increasing 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


$f \left({x}\right) \le f \left({\xi^-}\right) \le f \left({\xi}\right) \le f \left({\xi^+}\right) \le f \left({y}\right)$

provided that $a < x < \xi < y < b$.


$f$ is bounded above on $\left({a \,.\,.\, \xi}\right)$ by $f \left({\xi}\right)$.

By Limit of Increasing Function, the supremum is $f \left({\xi^-}\right)$.

So it follows that:

$\forall x \in \left({a \,.\,.\, \xi}\right): f \left({x}\right) \le f \left({\xi^-}\right) \le f \left({\xi}\right)$

A similar argument for $\left({\xi \,.\,.\, b}\right)$ holds for the other inequalities.