Question

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

Answer 1

Answer with Step-by-step explanation:

We are given that a function is a continuous on R

f:R$\rightarrow$R

We 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

$f^{-1}(R-H)=f^{-1}(R)-f^{-1}(H)=R-f^{-1}(H)$=Open set

When function is continuous then inverse image of open set is open

Hence, $f^{-1}(H)$is a closed set

Conversely,

Let inverse image of closed set H is closed

If H is closed set then R-H is open set

$f^{-1}(R-H)=f^{-1}(R)-f^{-1}(H)=R-f^{-1}(H)$

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.