Limit of Function in Interval

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

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.15 \ (4)$