Unformatted text preview: realvalued functions deFned on subsets of R 3 . ±irst, we need to set up some terminology. Defnition 3.6.3. A set S ⊂ R 3 of points in R 3 is closed if for any convergent sequence s i of points in S , the limit also belongs to S : s i → L and s i ∈ S for all i ⇒ L ∈ S. It is an easy exercise (Exercise 9 ) to show that each of the following are examples of closed sets: 1. closed intervals [ a,b ] in R , as well as halfclosed intervals of the form [ a, ∞ ) or ( −∞ ,b ]; 2. level sets L ( g,c ) of a continuous function g , as well as sets deFned by weak inequalities like { x ∈ R 3  g ( x ) ≤ c } or { x ∈ R 3  g ( x ) ≥ c } ;...
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 Fall '08
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 Calculus, Topology, Continuous function, Metric space, Extreme Value Theorem

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