The intermediate value theorem As a generalization of the above theorem we have

The intermediate value theorem as a generalization of

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Appendix. Some problems discussed in the class 5.3.1. True or false: (a) Let D be a compact subset of R and suppose that f : D R is continuous. Then f ( D ) is compact. 2 By “interval”, it means ( a, b ) , ( a, b ] , [ a, b ) or [ a, b ]. 3 To prove this, observe: p ( x ) = x 3 + ax 2 + bx + c = x 2 x + ( a + b x + c x 2 ) + as x + . Similarly, as x → -∞ , p ( x ) → -∞ . 108
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(b) Suppose thatf:DRis continuous. Then, there exists a pointx1inDsuch thatf(x1)f(x)for allxD.(c) LetDbe a bounded subset ofRand suppose thatf:DRis continuous.Thenf(D)is bounded. Solution:
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  • Fall '08
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  • Topology, Metric space, Compact space, image f, interval D

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