Continuous Inverse Theorem

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 $5$