Unformatted text preview: 3: Let f : [ a,b ] → R be continuous. Show the followings: a) For any ² > 0, there is a δ > 0 such that if x, ˜ x ∈ [ a,b ] with | x-˜ x | < δ , then | f ( x )-f (˜ x ) | < ² . b) Give an example of continuous f : (0 , 1) → R for which the statement in part (a) fails. Justify your answer. 4: Let f : (0 , 1) → R be increasing and bounded. Show that lim x → f ( x ) exists. Important results: Archimedean property, monotone and bounded sequences are conver-gent, closed and bounded sets are compact, Extreme Value Theorem, Intermediate Value Theorem, continuous maps on compacta are uniformly continuous. 1...
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- Fall '10
- Calculus, Continuous function, Order theory, Compact space