Originally Posted by

**SlipEternal** You did not show that the function is not continuous at $\displaystyle (0,0)$, but you did show that the derivative does not exist. You mixed up the subscripts for your $\displaystyle u$'s. The result when $\displaystyle u_2 \neq 0$ should be $\displaystyle \dfrac{u_2^2}{u_1}$. Your function was $\displaystyle g(x,y)$, not $\displaystyle f(x,y)$. Also, I don't understand the notation: $\displaystyle D_0f((u_1,0)) = 0$. To me, that reads the derivative with respect to zero at the point $\displaystyle (u_1,0)$. Since zero is a constant, not a variable, I don't think that derivative is well-defined. Once you show that the limit changes as you approach $\displaystyle (0,0)$ from different directions, you have sufficiently showed that the derivative does not exist at $\displaystyle (0,0)$.