Problem 1243 (difficulty: 3/10)

Prove that \(\displaystyle \displaystyle f(x,y)=\frac{x^3}{x^2+y^2}\), \(\displaystyle f(0,0)=0\) has directional derivatives at the origin in all directions. Is there a vector \(\displaystyle a\) such that for all \(\displaystyle v\) unit vector one has \(\displaystyle D_vf(0,0)=a\cdot v\)?

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