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21-356 Lecture 5

# F all continuous functions f u r whose partial

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Unformatted text preview: s di!erentiable at (0, 0) but the partial derivatives are not continuous at (0, 0). Deﬁnition 18 Given an open set U " RN , we denote by C 1 (U ) the space of !f all continuous functions f : U # R whose partial derivatives ! xi , 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 ! xi , 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 \$ RN , with x != y, let S be the segment of endpoints x and y, that is, S = {tx + (1 % t) y : t \$ [0, 1]} , and let f : S # R be such that f is continuous in S and there exists the !f directional derivative ! 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 (!x + (1 % !) y) &x % y& . "v (5) Lemma 21 Under the hypotheses of the previous theorem, the function g (t) := f (tx + (1 % t) y), t \$ [0, 1], is di!erentiable for all t \$ (0, 1), with g # (t) = "f (tx + (1 % t) y) &x % y& . "v Proof. Fix t0 \$ (0, 1) and consider g (t) % g (t0 ) f (tx + (1 % t) y) % f (t0 x + (1 % t0 ) y) = t % t0 t % t0 \$ % f t0 x + (1 % t0 ) y+ (t % t0 ) &x % y& "x!y" % f (t0 x + (1 % t0 ) y) x!y = &x % y&...
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