9. Let V be open in Rn, a E V, and 1 : V --t Rm. (a)

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9. Let V be open in Rn, a E V, and 1 : V --t Rm.

(a) Prove that Duf

(a) exists for u = ek if and only if fXk

(a) exists, in which case

(b) Show that if f has directional derivatives at a in all directions u, then the first-order partial derivatives of f exist at

a. Use Example 11.11 to show that the converse of this statement is false.

(c) Prove that the directional derivatives of (x, y) -I- (0,0)
(x, y) = (0,0)
exist at (0,0) in all directions u, but f is neither continuous nor differentiable at (0,0).

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An Introduction To Analysis

ISBN: 9780131453333

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Authors: William R. Wade

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9. Let V be open in Rn, a E V, and 1 : V --t Rm. (a)

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An Introduction To Analysis

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