Continuous Inverse Theorem

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Theorem

Let $f$ be a real function defined on an interval $I$.

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

Let $g$ be the inverse mapping to $f$.

Let $J := f \left[{I}\right]$ be the image of $I$ under $f$.


Then $g$ is strictly monotone and continuous on $J$.


Proof

From Strictly Monotone Real Function is Bijective, $f$ is a bijection.

From Inverse of Strictly Monotone Function, $f^{-1} : J \to I$ exists and is strictly monotone.

From Surjective Monotone Function is Continuous, $f^{-1}$ is continuous.


Hence the result.

$\blacksquare$


Sources