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Monotone and antitone functions
(not over ℝ just the domain you see =
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.,
lim f(x) = β≤+∞ and limf(x ) = α≥-∞, if f is strictly decreasing,
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.,
lim(f -1)(y)=b and lim(f -1 )(y)=a, if f is strictly decreasing.
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.
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