21-356 Lecture 4

21-356 Lecture 4

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Unformatted text preview: ctional derivatives of f at x0 exist and that ! v (x0 ) = )N ! f i=1 ! xi (x0 ) vi for every direction v. Prove that f is di!erentiable at x0 . (ii) Assume that all the partial derivatives of f at x0 exist, that the directional !f derivatives ! v (x0 ) exist for all v ! S , where S is dense in the unit sphere, )N ! f !f and that ! v (x0 ) = i=1 ! xi (x0 ) vi for every direction v ! S . Prove that f is di!erentiable at x0 . The next theorem gives an important su"cient condition 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 functio...
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This document was uploaded on 03/31/2014.

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