cal7 - Chapter Seven Continuity Derivatives and All That...

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7.1 Chapter Seven Continuity, Derivatives, and All That 7.1 Limits and Continuity Let x R 0 n and r > 0. The set B r r ( ; ) { :| | } a x R x a n = - < is called the open ball of radius r centered at x 0 . The closed ball of radius r centered at x 0 is the set B r r { :| | } a x R x a n = - Now suppose D R n . A point a D is called an interior point of D if there is an open ball B r ( ; ) a D The collection of all interior points of D is called the interior of D, and is usually denoted int D . A set U is said to be open if U = int U . Suppose f : D R p , where D R n and suppose a R n is such that every open ball centered at a meets the domain D . If y R p is such that for every ε > 0, there is a δ > 0 so that | ( ) | f x y - < ε whenever 0 < - < | | x a δ , then we say that y is the limit of f at a . This is written lim ( ) x a x y = f , and y is called the limit of f at a Notice that this agrees with our previous definitions in case n = 1 and p =1,2, or 3. The usual properties of limits are relatively easy to establish: lim( ( ) ( )) ( ) ( ) x a x a x a x x x x + = + f g f g , and ( ) ( ) x a x a x x = af a f . Now we are ready to say what we mean by a continuous function f : D R p , where D R n . Again this definition will not contradict our previous lower dimensional
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7.2 definitions. Specifically, we say that f is continuous at a D if lim ( ) ( ) x a x a = f f . If f is continuous at each point of its domain D , we say simply that f is continuous . Example Every linear function is continuous. To see this, suppose f : R R n p is linear and a R n . Let ε > 0. Now let M f f f = max{| ( )|,| ( )|, ,| ( )|} e e e 1 2 n K and let δ ε = nM . Then for x such that 0 < - < | | x a δ , we have | ( ) ( )| | ( ) ( )| |( ) ( ) ( ) ( ) ( ) ( )| | || ( )| | || ( )| | || ( )| (| | | | | |) f f f x x x f a a a x a f x a f x a f x a f x a f x a f x a x a x a M n n n n n n n n n n n n x a e e e e e e e e e e e e - = + + + - + + + = - + - + + - - + - + + - - + - + + - 1 1 2 2 1 1 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 1 2 2 K K K K K - < n M | | x a ε Thus ( ) ( ) x a x a = f f and so f is continuous. Another Example Let f : R R 2 be defined by f f x x x x x x x x ( ) ( , ) , , x = = + + 1 2 1 2 1 2 2 2 1 2 2 2 0 0 for otherwise . Let’s see about ( ). ( , ) x x 00 f Let x = α ( , ) 11 . Then for all α ≠ 0 , we have f f ( ) ( , ) x = = + = α α α α α 2 2 2 1 2 .
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7.3 Now. let x = = α α ( , ) ( , ) 10 0 . It follows that all α ≠ 0 , f ( ) x = 0 What does this tell us? It tells us that for any δ > 0 , there are vectors x with 0 00 < - < | ( , )| x δ such that f ( ) x = 1 2 and such that f x = 0 This, of course, means that lim ( ) ( , ) x x f does not exist. 7.2 Derivatives Let f : D R p , where D R n , and let x D 0 int Then f is differentiable at x 0 if there is a linear function L such that | | [ ( ) ( ) ( )] h 0 0 0 h x h x L h 0 + - - = 1 f f . The linear function L is called the derivative of f at x 0 . It is usual to identify the linear function L with its matrix representation and think of the derivative at a p n × matrix. Note that in case n = p = 1, the matrix L is simply the 1 1 × matrix whose sole entry is the every day grammar school derivative of f .
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This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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cal7 - Chapter Seven Continuity Derivatives and All That...

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