Strictly Monotone Real Function is Bijective
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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.
From Strictly Monotone Mapping with Totally Ordered Domain is Injective, $f$ is an injection.
From Surjection by Restriction of Codomain, $f: I \to J$ is a surjection.
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.9$