Class Notes (810,512)
Canada (494,140)
Statistics (237)
STAT312 (22)

Limits & continuity.pdf

4 Pages
Unlock Document

University of Alberta
Douglas Wiens

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
More Less

Related notes for STAT312

Log In


Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.