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Unformatted text preview: Lecture 12 Definition 1. Let U be open in X . When F : U Y is differentiable at all points a U , its derivative is the map F : U L ( X,Y ) : a 7 F a and we call F continuously differentiable when F is continuous. Theorem 1. Suppose U contained in X 1 X 2 ; then F : U Y is continu ously differentiable if and only if 1 F and 2 F are defined and continuous at each point of U . Proof. Scribal Exercise For any a = ( a 1 ,a 2 ) U , let  a  = max { a 1  ,  a 2 } . ( ) First, suppose that F is continuously differentiable. This means that F a is defined for all a U and that F is continuous at a . By the last theorem of Lecture 9 1 F a and 2 F a exist at every a U and F a ( h 1 ,h 2 ) = 1 F a ( h 1 ) + 2 F a ( h 2 ) . Specifically, 1 F a ( h 1 ) = F a ( h 1 , 0) 2 F a ( h 2 ) = F a (0 ,h 2 ) . Now consider the maps defined below: 1 : X 1 X 1 X 2 : h 1 7 ( h 1 , 0) 2 : X 2 X 1 X 2 : h 2 7 (0 ,h 2 ) . 1 1 and 2 are linear and k 1 ( h 1 ) k = k ( h 1 , 0) k = k h 1 k k 2 ( h 2 ) k = k (0 ,h 2 ) k = k h 2 k . It follows that k 1 k = 1 = k 2 k . We can see that 1 F : U L ( X 1 ,Y ) : a 7 F a 1 2 F : U L ( X 2 ,Y ) : a 7 F a 2 . Since F is continuous at each a let > 0 and choose such that if b U with  a b  < then k F a F b k < , thus for i = 1 or 2 k i F a i F b k = k F a i F b i k = k ( F a F b ) i k k F a F b kk i k = k F a F b k < ....
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.
 Summer '10
 Staff
 Algebra, Derivative

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