Class Notes (838,846)
Statistics (248)
STAT312 (22)
Lecture

Continuity and Differentiation.pdf

5 Pages
73 Views

School
Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
Fall

Description
76 14. Continuity and Di erentiation  Suppose maps x R to a real num- ber (x). We write : R R; is the domain of . Recall the denition: The function is continuous at a pointa if, as x a , we have that (x) (a) (equivalently, | (x) a )| 0 as kx ak 0). 2  Example: ( ) = , = (0 ). Then if 0 we have 0 | ( ) ( )| = | || + 2 | | | · (| | + 2 )| 0 as | | 0. Here we used the triangle inequality: | + | | | + | | 77  Inmum and suprema (think min and max, but ... ): For any set , is a lower bound if for all . If there is a nite lower bound then there are many; the largest of them is the greatest lower bound ( ) or inmum (inf). Otherwise the inf is . Similarly with upper bound, least upper bound ( ) or supremum (sup).  Here is a simple but very useful result (Lab 5). If { } is increasing: and +1 bounded above: , then = sup is nite (why?) and as .  Further properties of continuous functions : R R :  If is continuous on a closed and bounded set then it is bounded there. Thus the inf and sup are nite, and are attained: there are points with ( ) ( ) ( ) for a
More Less

Related notes for STAT312
Me

OR

Join OneClass

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

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.