Differentiability in $f:\mathbb{R}^2 \to \mathbb{R}^2$ v/s $g:\mathbb{C} \to \mathbb{C}$ I was reading around stuff on differentiability in $\mathbb{C}$ and wondered whether it is same as differentiability in $\mathbb{R}^2$. I approached a professor and gave me an example and asked me to think over it.
$f:\mathbb{R}^2 \to \mathbb{R}^2$,   $ f(x,y)=(x,-y)$
And $g:\mathbb{C} \to \mathbb{C}$,   $ g(z)=\bar z$,       the conjugate of z
If we could plot both the graphs, both would be exactly the same.
Clearly $g$ is not differentiable anywhere in $\mathbb{C}$
Also $f$ is a Linear Transformation (as we can find a Matrix of transformation for the given function) and hence differentiable.
Why is there such a difference.
I read Complex differentiability vs differentiability in $\mathbb{R}^2$ and couldn't understand much as I'm just a beginner in Linear Algebra. 
Please, try to explain in the simplest way possible. Thanks.
 A: This is not so much a matter of limits, as a matter of linear algebra.
A general linear map
$$A:\quad{\mathbb R}^2\to{\mathbb R}^2,\quad{\bf z}\to{\bf w}=A{\bf z}\ ,$$
being a map between real-two-dimensional spaces,
appears in coordinates $(x,y)$, resp. $(u,v)$, as
$$\left.\eqalign{u&=a_{11}x+a_{12}y\cr v&=a_{21}x+a_{22}y\cr}\right\}\ .$$
Here $\bigl[a_{ik}\bigr]=:\bigl[A\bigr]$ can be an arbitrary $(2\times2)$-matrix with real entries. In particular, the derivative $d{\bf f}({\bf p})$ of a map $$f:\quad{\mathbb R}^2\to{\mathbb R}^2$$ at a point ${\bf p}$ can be any such $A$, and therefore can have $4$ independently chosen entries in its matrix.
Contrasting this, a linear map
$$T:\quad{\mathbb C}\to{\mathbb C},\quad z\to w=Tz\ ,$$
being a map between complex-one-dimensional spaces, appears in the complex coordinates $z\ (=x+iy)$, resp. $w\ (=u+iv)$, as
$$w=\lambda\>z,\qquad\lambda=\mu+i\nu\in{\mathbb C}\ .\tag{1}$$
Separating real and imaginary parts in $(1)$ we have
$$\left.\eqalign{u&=\mu x-\nu y\cr v&=\nu x+\mu y\cr}\right\}\ .$$
The "geometric" map $T$, which appeared as one-dimensional complex-linear map in $(1)$, is now two-dimensional real linear with matrix
$$\bigl[T\bigr]=\left[\matrix{\mu&-\nu\cr \nu&\mu\cr}\right]\ .\tag{2}$$
The matrix appearing here is not an arbitrary real $(2\times2)$-matrix, but has a particular structure with only $2$ "real degrees of freedom". That the $\bigl[T\bigr]$ in $(2)$ has this form is the content of the famous CR differential equations for analytic functions of $z$.
