Appendix. Some problems discussed in the class5.3.1. True or false:(a) LetDbe a compact subset ofRand suppose thatf:D→Ris continuous.Thenf(D)is compact.2By “interval”, it means (a, b),(a, b],[a, b) or [a, b].3To prove this, observe:p(x) =x3+ax2+bx+c=x2x+ (a+bx+cx2)→+∞asx→+∞. Similarly,asx→ -∞,p(x)→ -∞.108
(b) Suppose thatf:D→Ris continuous. Then, there exists a pointx1inDsuch thatf(x1)≥f(x)for allx∈D.(c) LetDbe a bounded subset ofRand suppose thatf:D→Ris continuous.Thenf(D)is bounded.Solution:
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Topology,Metric space,Compact space,image f,interval D