Can you prove that def=hgf? Justify your answer.
1 answer:
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Answer with Step-by-step explanation:
We are given that a function is a continuous on R
f:RR
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
=Open set
When function is continuous then inverse image of open set is open
Hence, 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
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.