I came across the following problem:
Question: Let \begin{equation*} f : \mathbb{R}^2 \to \mathbb{R}, ~~~ (x,y) \mapsto \begin{cases} 0, & (x,y) = (0,0) \\ \frac{x^3}{x^2 + y^2}, & (x,y) \neq (0,0). \end{cases} \end{equation*} Show that $f$ is partially differentiable everywhere, and determine where $g$ is differentiable.
And the answer (some detail omitted):
Answer: Let $\mathbf{v} = (h,k) \neq (0,0)$. Then the directional derivatives are \begin{equation*} D_{\mathbf{v}}f(\mathbf{0}) = \displaystyle \lim_{t \to 0} \frac{f(\mathbf{0} + t\mathbf{v}) - f(\mathbf{0})}{t} = \displaystyle \lim_{t \to 0} \frac{t^3h^3}{t^3(h^2 + k^2)} = \displaystyle \lim_{t \to 0} \frac{h^3}{h^2 + k^2} = \begin{cases} h, & k=0 \\ \frac{h^3}{h^2 + k^2}, & k \neq 0. \end{cases} \end{equation*} For $\mathbf{v} = \mathbf{e_1} = (1,0)$ and $\mathbf{v} = \mathbf{e_2} = (0,1)$ the partials at the origin are $\frac{\partial f}{\partial x}(0,0) = 1$ and $\frac{\partial f}{\partial y}(0,0) = 0$. For everywhere else, $f$ is rational and the partials exists, and specifically we define them as \begin{equation*} \frac{\partial f}{\partial x} = \begin{cases} \frac{x^4 + 3x^2y^2}{(x^2 + y^2)^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} ~~~~~~~\text{and}~~~~~~~ \frac{\partial f}{\partial y} = \begin{cases} \frac{-2x^3y}{(x^2 + y^2)^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0). \end{cases} \end{equation*} The function is differentiable everywhere except at the origin, since if we consider $\mathbf{x}_n = \left(\frac{1}{n}, \frac{1}{n}\right) \to 0$ then \begin{equation*} f(\mathbf{x}_n) = \frac{1}{n^3} \cdot \frac{n^2}{2} = \frac{n^2}{2n^3} = \frac{1}{2n} \to \infty, \end{equation*} as $n \to 0$.
My question is can't we take from the fact the the partials at the origin are different for $\mathbf{e}_1$ and $\mathbf{e}_2$, i.e. $\frac{\partial f}{\partial x}(0,0) \neq \frac{\partial f}{\partial y}(0,0)$, that it is not continuous at the origin. The example at the end is correct, but is it necessary if we can simply see that the partials are different at the origin?
