Strictly Monotone Function is Bijective

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Theorem

Let $f$ be a real function which is defined on $I \subseteq \R$.

Let $f$ be strictly monotone on $I$.

Let the image of $f$ be $J$.


Then $f: I \to J$ is a bijection.


Proof

From Strictly Monotone Mapping is Injective, $f$ is an injection.

From Surjection by Restriction of Codomain, $f: I \to J$ is a surjection.

$\blacksquare$


Sources