Continuity preserves closedness and boundedness
A continuous function on a closed bounded set has a bounded closed image: is bounded and closed whenever is continuous and is.
Proposition 2.20 (Continuity Preserves Closedness and Boundedness)
Proof
Let be closed and bounded. If is continuous, then is a bounded closed subset of .
Boundedness. If were unbounded, choose with . Then is bounded since is, so by Bolzano-Weierstrass a subsequence converges to some ; closedness of gives . But , so cannot converge as , contradicting continuity of at . Hence is bounded.
Closedness. Take in converging to , and pick with . The sequence lies in the bounded set , so copying the argument above yields a subsequence . By continuity of ,
hence .
Related
Stated in
- Proposition 2.20 (Continuity Preserves Closedness and Boundedness)§2.3 Extreme Value Theorem