This exercise shows that there exists a function that is not differentiable at (0, 0) even though all directional derivatives at (0, 0) exist Define (x, y) x 2 y (x 2 y 2 ) for (x, y) 0 and (0, 0) 0 (a) Use the limit definition to show that D v (0, 0) exists for all vectors v Show that x(0, 0) y(0,...

a By the limit definition and since f0 0 0 we have fx0 0 li ...

This exercise shows that there exists a function that is not differentiable at (0, 0) even though

Question:

This exercise shows that there exists a function that is not differentiable at (0, 0) even though all directional derivatives at (0, 0) exist. Define ƒ(x, y) = x²y/(x² + y²) for (x, y) ≠ 0 and ƒ(0, 0) = 0.
(a) Use the limit definition to show that D_vƒ(0, 0) exists for all vectors v. Show that ƒx(0, 0) = ƒy(0, 0) = 0.
(b) Prove that ƒ is not differentiable at (0, 0) by showing that Eq. (7) does not hold.

f(x, y) lim (x,y) (0,0) x + y =0