Inverse Function Theorem
If is continuous and strictly monotone, then with , is a bijection, and is continuous and strictly monotone.
Version 2
Let be continuous on and differentiable on . Assume that for all . Then is bijective with inverse continuous and differentiable on with
Proof
A continuous, strictly monotone function maps bijectively onto , and the inverse is again continuous and strictly monotone.
Bijectivity. Monotonicity forces the extreme values of on to occur at the endpoints, so maps into ; the Intermediate Value Theorem makes it surjective onto , and strict monotonicity makes it injective.
Monotonicity of the inverse. WLOG let be strictly increasing. If were not, there would be in with ; applying the increasing function gives , a contradiction. Hence is strictly increasing.
Continuity of the inverse. Fix and .
-
If , choose with . Strict increase gives , and satisfies
-
If , take and ; then with forces , so .
-
The case is similar.
Related
Stated in
- Proposition 2.25 (Inverse Function Theorem, Version 1)§2.4 Intermediate Value Theorem
- Theorem 3.13 (Inverse Function Theorem, Version 2)§3.2 Mean Value Theorems