Let f: R --> R Prove that f is continuous on R iff f-1(H) is a closed set whenever H is a closed set
Accepted Solution
A:
Answer with Step-by-step explanation:We are given that a function is a continuous on Rf:R[tex]\rightarrow [/tex]RWe have to prove that if function is continuous ton R iff inverse image of closed set H is closed.Let H be a closed set and function is continuous then R-H is a opens set [tex]f^{-1}(R-H)=f^{-1}(R)-f^{-1}(H)=R-f^{-1}(H)[/tex]=Open set When function is continuous then inverse image of open set is openHence, [tex]f^{-1}(H) [/tex]is a closed set Conversely,Let inverse image of closed set H is closedIf H is closed set then R-H is open set [tex]f^{-1}(R-H)=f^{-1}(R)-f^{-1}(H)=R-f^{-1}(H)[/tex]When inverse image of closed set is closed then R-inverse image of H is opens set When inverse image of open set is open then the function is continuous.Hence, function is continuous.Hence proved.