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Lecture

Continuity and Differentiation.pdf

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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
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