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21-356 Lecture 6 - Wednesday The next exercise shows that...

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Definition 18 Given an open set U " R N , we denote by C 1 ( U ) the space of all continuous functions f : U # R whose partial derivatives ! f ! x i , i = 1 , . . . , N , exist at all x $ U and are continuous. Remark 19 The hypothesis that f is continuous is redundant. It is enough to define C 1 ( U ) the space of all functions f : U # R whose partial derivatives ! f ! x i , i = 1 , . . . , N , exist at all x $ U and are continuous. Indeed, for any such function f , by Theorem 16, we have that f must be di ! erentiable at all x $ U , and, in turn, by Theorem 9, f must be continuous. Next we extend the mean value theorem to functions of several variables. Theorem 20 (Mean Value Theorem) Let x , y $ R N , with x ! = y , let S be the segment of endpoints x and y , that is, S = { t x + (1 % t ) y : t $ [0 , 1] } , and let f : S # R be such that f is continuous in S and there exists the directional derivative ! f ! v ( z ) for all z $ S except at most x and y , where v := x ! y " x ! y " . Then there exists ! $ (0 , 1) such that f ( x ) % f ( y ) = " f " v ( ! x + (1 % ! ) y ) & x % y & . (5) Lemma 21 Under the hypotheses of the previous theorem, the function g ( t ) := f ( t x + (1 % t ) y ) , t $ [0 , 1] , is di ! erentiable for all t $ (0 , 1)
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