Partial derivative and complex derivative

Question

In my Complex Analysis class I had the following question, I can't seem to wrap my head around:

Let $z=x+iy$ and $x,y\in \mathbb{R}; f: \mathbb{C} \to \mathbb{C}$ and $$ f(x+iy)=\frac{xy^2(x+iy)}{x^2+y^4}$$ for $z\neq0$ and $0$ otherwise. And I was supposed to prove the following:

1. $f$ is in $z_0=0$ partially differentiable. I thought it's pretty easy to show using $\lim_{h \mapsto 0} \frac{f(h)-f(0)}{h}=\lim_{h \mapsto 0} \frac{f(ih)-f(0)}{h}$. I get $0$ for both.

2. Now I'm supposed to show the Cauchy-Rieman-equations hold for $z_0$. With my solution from 1.) I said they hold it, since they are both equal to zero.

3. Now we are supposed to show the existence of $f_{\phi}(0)=\lim_{t\rightarrow 0}\frac{f(te^{i\phi})-f(0)}{t}$ and is independent of the choice for $\phi$. Using Wolframalpha I saw, that it converges to $0$ which would be obviously independent of the choice of $\phi$. There were lots of calculations and I think I got lost in between somewhere.

4. $f$ is not complex differentiable in $z_0$. This is where I got stuck. Since the partial derivatives (according to my solution) are both equal to zero, the $\frac{\partial f}{\partial \overline{z}}=0$. I guess it shouldn't be zero but I can't seem to find my mistake. Hopefully some of you can help me with this.

Answer 1

There are different notions of differentiability at play here, and I think realising this is the point of the exercise: "although we have CR equations, and the limit to zero along every line coinciding, the function is not complex differentiable there"

1. Partial differentiable at points: existence of these limits you wrote in 1 along real/imaginary axes (as you proved at z_0)
2. Cauchy-Riemann at points: we have partial derivatives at the point in question, and they satisfy the CR-equations (as you said hold at z_0)
3. Real differentiable at points: we have a "linear approximation" at that point. This implies existence of partial derivatives in the point. However the link provides two examples of partial differentiable functions at a point, that are not real differentiable.
4. Complex differentiable at points: the limit of the difference quotient at that point along any subset of $C$ (not just the lines, as you computed in 3) exists

(5. holomorphic at points: complex differentiable in an open around the point, this is the one related to complex analyticity)

The condition you wrote down in 4 is the same as the CR-equations, so this will be 0 at 0. However, complex differentiable at a point implies real differentiable at a point: this is must hold if we want our function to be complex differentiable.

The point is this: complex differentiable is the same as real differentiable + CR-equations. Hope this helps.