Strictly Monotone Real Function is Bijective

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 with Totally Ordered Domain is Injective, $f$ is an injection.

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

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.9$