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Lecture

# Limits & continuity.pdf

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School
University of Alberta
Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
Fall

Description
72 13. Limits & continuity  Open and closed sets in R ; limits:  Neighbourhood of a point a , of radius : (a ) = {x| |x a|| }  R is open if a (a) for all su ciently small 0. Example: (0 1).  A sequence { x } tends to a poina: x a as ifx gets arbitrarily close ao as gets larger and larger. More formally, any neighbourhood of a , no matter how small, will eventually contaix from some point on- ward. More formally yet, for any radius , we can nd an large enough that, once , 73 all of thx lie in a ). This required will typically get larger asgets smaller. Finally, = ( )( x (a)) read for all there exists an , that depends on , such that implies thatx (a). Equivalently (why?): x a |x a || 0. This is for a nite; obvious modications otherwise. You should derive an appropri- ate denition of   ( scalars, not vectors). Example = 1 1 1 as .  A point a is a limit point of R if there is a sequence {x } such thatx a . Example = (0 1) = 1 1; = 1 ( ) 74  R is closed if it contains all of its limit points. Examples = [0 1] (if 0 or 1 it cannot be the limit of a sequence in [0 1]).  A set
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