The existence theorem of inverse function of mathematical analysis, continuity theorem and derivation theorem

The existence theorem of inverse function

If the function y=f (x), x∈df y = f (x), x \in D_f is strictly monotonically increasing (decreasing), then there is its inverse function
X=f−1 (y): rf→x x = f^{-1} (y): R_f \rightarrow X, and f−1 (y) f^{-1} (Y) is also strictly monotonically increased (reduced). Proof:

It is advisable to set y=f (x), x∈df y = f (x), x \in d_f strictly monotonic increments, ∀x1,x2∈df,x1<x2⇒f (x1) <f (x2) \forall x_1, x_2 \in d_f, x_1, so ∀x1,x2 ∈df,f (x1) =f (x2) ⇒x1=x2 \forall x_1, x_2 \in D_f, f (x_1) = f (x_2) \rightarrow x_1 = x_2, so there is an inverse function f−1 (y), Y∈rf f^{-1} (y), y \i N R_f.
∀y1,y2∈df−1