Answer to Question #177918 in Real Analysis for Nikhil Singh

Question

Prove or disprove the following statement

‘ Every strictly increasing onto function is invertible'

Answer 1

Let F: R $\to$ R be any strictly increasing into function.

We show that F is bijective.

To this end, let x,y $\in$ R with x $\ne$ y then so we assume x<y, so that F(x)<F(y) implies F(x) $\ne$ F(y). That is f is injective, and since F is onto by the hypothesis, it is bijective and thus Invertible.