21-356 Lecture 5

Theorem 16 let e rn let f e r let x0 e assume

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Unformatted text preview: for di"erentiability at a point x0 . Theorem 16 Let E ' RN , let f : E # R, let x0 & E " . Assume that there ex!f ists r > 0 such that B (x0 , r) ' E and the partial derivatives ! xj , j = 1, . . . , N , exist for every x & B (x0 , r) and are continuous at x0 . Then f is di!erentiable at x0 . Proof. Let x, y & B (x0 , r). Write x = (x1 , . . . , xN ) and y = (y1 , . . . , yN ). Then f (x) ! f (y) = (f (x) ! f (x1 , . . . , xN #1 , yN )) + · · · + (f (x1 , y2 , . . . , yN ) ! f (y)) . By the mean value theorem applied to the function of one variable f (x1 , . . . , xN #1 , ·), f (x) ! f (x1 , . . . , xN #1 , yN ) = 10 !f (zN ) (xN ! yN ) , ! xN where zN := (x1 , . . . , xN #1 , "N xN + (1 ! "N ) yN ) for some "N & (0, 1). Note that (zN ! y( " (x ! y( . Similarly, f (x1 , . . . , xi , yi+1 , . . . , yN )!f (x1 , . . . , xi#1 , yi , yi+1 , . . . , yN ) = !f (zi ) (xi ! yi ) , ! xi where zi := (x1 , . . . , xi#1 , "i xi + (1 ! "...
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This document was uploaded on 03/31/2014.

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