Question: Let V be open in Rn, a V, and f : V Rm. a) Prove that Duf(a) exists for u = ek if and only

Let V be open in Rn, a ˆˆ V, and f : V †’ 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 derivates of

exist at (0, 0) in all directions u, but f is neither continuous nor differentiable at (0, 0).

at Def(a) = (a). r2y f(x, y) = 1K4+y2 (x,y)(0, 0) (x, y) = (0, 0)

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