Limit of Function in Interval

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Theorem

Let $f$ be a real function which is defined on the open interval $\left({a \,.\,.\, b}\right)$.

Let $\xi \in \left({a \,.\,.\, b}\right)$

Suppose that, $\forall x \in \left({a \,.\,.\, b}\right)$, either:

$\xi \le f \left({x}\right) \le x$

or:

$x \le f \left({x}\right) \le \xi$


Then $f \left({x}\right) \to \xi$ as $x \to \xi$.


Proof

Note that $\left|{f \left({x}\right) - \xi}\right| \le \left|{\xi - x}\right|$.

From Limit of Absolute Value‎ we have that $\left|{x - \xi}\right| \to 0$ as $x \to \xi$.

The result follows from the Squeeze Theorem.

$\blacksquare$


Sources