I'm looking at a textbook example that shows the existence of partials/directional derivatives is insufficient to guarantee continuity.
The example function is: $$ f(x,y) = \cfrac{xy^2}{x^2+y^4}, x\neq 0$$ $$f(x,y) = 0, x = 0$$
For any $(u_1,u_2)$ in $\mathbb{R}^2$, The directional derivative of $f$ at $(0,0)$,$Df(0,0;u) = \lim_{\alpha \to 0} \cfrac{u_1 u_2^2}{u_1^2 + \alpha^2 u_2^4} = \cfrac{u_2 ^2}{u_1}$ if $u_1 \neq 0$ and $= 0$ if $u_1 =0$ .
Okay, great, so the directional derivative exists for all $u$.
The next part is what I don't get.
"$f$ has a value of $\frac{1}{2}$ at all points on the curve $x = y^2$ except at the origin, where it is zero. Hence, $f$ is not continuous at $(0,0)$."
First question - should we have seen this coming, given the fact that the directional derivative is defined as a hybrid function, conditional on where we are in the space?
Second question - it seems arbitrary to me why they chose the curve $x = y^2$ - can someone explain the intuition behind this choice? If I had to do the same confirmation for a random, unfamiliar function, how would I know to chose $x = y^n$ ?