Limit of Monotone Real Function/Increasing/Corollary

Corollary to Limit of Increasing Function

Let $f$ be a real function which is increasing on the open interval $\openint a b$.

If $\xi \in \openint a b$, then:

$\map f {\xi^-}$ and $\map f {\xi^+}$ both exist

and:

$\map f x \le \map f {\xi^-} \le \map f \xi \le \map f {\xi^+} \le \map f y$

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

Proof

$f$ is bounded above on $\openint a \xi$ by $\map f \xi$.

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By Limit of Increasing Function, the supremum is $\map f {\xi^-}$.

So it follows that:

$\forall x \in \openint a \xi: \map f x \le \map f {\xi^-} \le \map f \xi$

A similar argument for $\openint \xi b$ holds for the other inequalities.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.5$