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
- 2011: Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis (4th ed.): $\S 5.6$: Theorem $5$