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Lecture

# Limits & continuity.pdf

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University of Alberta

Statistics

STAT312

Douglas Wiens

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 modications
otherwise. You should derive an appropri-
ate denition 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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