Let f : R → R be a map of class C 1 . Show...

racodelitusmn

racodelitusmn

Answered

2022-07-07

Let f : R R be a map of class C 1 . Show the set P = { x R : f ( x ) 0 } is open.

Answer & Explanation

Caiden Barrett

Caiden Barrett

Expert

2022-07-08Added 20 answers

Step 1
The function f′ is continuous . So inverse image of closed sets is a closed set. as the singleton {0} is closed.
Step 2
R P = ( f ) 1 ( { 0 } ) is closed as f′ is continuous. So ( R ( f ) 1 ( { 0 } ) ) = P is open.

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