Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function defined by
is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function defined by
is not integrable.
a) False. The complement of the interval ]0, 1] in ℝ is (−∞, 0] ∪ (1, ∞), which contains closed endpoints 0 and 1. Since it includes its boundary points, it is not an open set. b) True. A bounded sequence need not be convergent. For example, the sequence __________ ________ ____ _________ ___ ___ ________ ___ ______ __________.
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_____ _________ _______ __________ __________ _________ ______ _____ _______ ___ ________.
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___ _________ ____ _________ __________ _____ __________ __________ ______ __________ _________.
_________ ______ ________ _______ _____ ______ ________ ________.
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_____ _________ _____ ____ ________ _____ _____ ______ _________ __________ __________ ______.
________ __________ ______ __________ ___ _________ __________.
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______ _______ ___ _________ ________ __________ _______ __________ ___ _________.
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__________ ______ ________ ___ ____ ____ ________.
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.
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