isomorphismes:

Monotone and antitone functions

(not over ℝ just the domain you see = `0<x<1⊂ℝ`

)

These are examples of invertible functions.

**Theorem on the inverse function of continuous strictly monotonic functions:**

Suppose the function f:(a,b)→ℝ with -∞≤a<b≤+∞ is strictly increasing (resp., decreasing) and continuous. Let

*lim f(x)=α≥-∞ and lim f(x)=β≤+∞ , if f is strictly increasing, resp.,*

*x→a+ x→b−*

*lim f(x) = β≤+∞ and limf(x ) = α≥-∞, if f is strictly decreasing,*

*x→a+ * x→b−

Then f maps the interval (a,b) invertibly onto the interval (α,β). The inverse function f ^{-1}:(α,β)→(a,b) is also strictly increasing (resp., decreasing) and continuous, and we have:

*lim(f *^{-1})(y)=a and lim(f ^{-1})(y)=b, if f is strictly increasing, resp.,

*y→α+ y→β− *

*lim(f *^{-1})(y)=b and lim(f ^{-1} )(y)=a, if f is strictly decreasing.

*y→α+ y→β−*

Analogous statements hold for semi-open or closed intervals [a,b].[Source]

Also, If (m,n) ⊂ (a,b), the function f:(a,b)→ℝ, that is also true.