Inverse of Strictly Monotone Function

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 $f$ always has an inverse function $f^{-1}$ and:
 * if $f$ is strictly increasing then so is $f^{-1}$
 * if $f$ is strictly decreasing then so is $f^{-1}$.

Proof
The function $f$ is a bijection from Strictly Monotone Real Function is Bijective.

Hence from Bijection iff Inverse is Bijection, $f^{-1}$ always exists and is also a bijection.

From the definition of strictly increasing:
 * $x < y \iff \map f x < \map f y$

Hence:
 * $\map {f^{-1} } x < \map {f^{-1} } y \iff \map {f^{-1} } {\map f x} < \map {f^{-1} } {\map f y}$

and so:
 * $\map {f^{-1} } x < \map {f^{-1} } y \iff x < y$

Similarly, from the definition of strictly decreasing:
 * $x < y \iff \map f x > \map f y$

Hence:
 * $\map {f^{-1} } x < \map {f^{-1} } y \iff \map {f^{-1} } {\map f x} > \map {f^{-1} } {\map f y}$

and so:
 * $\map {f^{-1} } x < \map {f^{-1} } y \iff x > y$