Inverse of Strictly Monotone Continuous Real Function is Strictly Monotone and Continuous

Theorem

Let $f$ be a continuous real function which is defined on the closed interval $I := \closedint a b$.

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

Then $f$ has an inverse function $f^{-1}$ which is continuous and strictly monotone on $f \sqbrk I$.

Proof

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

From Inverse of Strictly Monotone Function, $f^{-1}$ is strictly monotone on on $f \sqbrk I$.

From Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone, $f^{-1}$ is a continuous real function.

Hence the result.

$\blacksquare$

Examples

Arbitrary Example

Consider the real function:

$\forall x \in \closedint 0 1: \map f x = y = 2 x + 3$

This has an inverse:

$\map {f^{-1} } y = x = \dfrac {y - 3} 2$

on the closed interval $\closedint 3 5$

Hence we can say:

$f: x \mapsto 2 x + 3$ on $\closedint 0 1$

and:

$f^{-1}: x \mapsto \dfrac {x - 3} 2$ on $\closedint 3 5$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse: 1. (of a function)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse: 1. (of a function)