Answer to Question #310437 in Real Analysis for Nikhil Singh

Question

Prove or disprove the following statement

‘ Every strictly increasing onto function is invertible'

Answer 1

$Solution: \\Let ~F:R \rightarrow R~be ~any~strictly~increasing~into~function. \\We~show~that~F~is~bijective. \\To~this~end~,let~x,y\in R~with~x \neq y~then~so~we~assume~x<y,so~that~ \\F(x)<F(y)~ implies~F(x)\neq F(y).That~is~f ~is ~ injective~and~since~F ~is~onto~by~the~hypothesis, \\it~is~bijective~and~thus~invertible.$