$F(x,y)=(x^2,y)$ is not differentiable (According to WA), what am I missing?

Question

Either wolfram alpha is getting crazy or I just completely misunderstood differentiability.

Shouldn't it be enough for the partial derivatives to exist and to be continuous? Would the differential not just be the jacobian? So, it should be differentiable...but is not?

PS: This is part of a wider question that goes $f(z)= x^2+iy$. Am I correct with the corresponding $F$ in $\mathbb{R}^2$?

Answer 1

Differentiability in $\mathbb{C}$ is not the same as in $\mathbb{R}^2$. Differentiable complex functions must satisfy the Cauchy-Riemann equations, which is not the case of $f(z)=x^2+i y$.