First, nice work so far. Looking to the Cauchy-Riemann Equations is a good idea.
To answer your first question,this is not enough. Satisfying only one of the Cauchy-Riemann Equations is NOT enough to prove differentiability. Satisfying BOTH equations at a point is part of what is needed to deduce differentiability. We will further discuss what is needed to conclude differentiability at a point below.so for
, which is true.
Is this enough to show that it is diffentiable?
You used the term "analytic" in your previous post, have you seen the definition for holomorphic before?
A function is said to be holomorphic at if it is differentiable at every point in neighborhood of .
The key here is that the function is differentiable not just at , but at EVERY point in some neighborhood around .
Now, it turns out that a function is holomorphic at a point if and only if it is analytic at that point.
So, if we can show that your function is not holomorphic anywhere, then (by the above if and only if) it is not analytic anywhere. This is what we will outline below.
The Cauchy-Riemann Equations are a necessary condition for complex differentiability; meaning that if is differentiable at , then satisfies the Cauchy-Riemann Equations at .
However, there is a partial converse to the previous statement. It says that:
If and are differentiable at and they satisfy the Cauchy-Riemann Equations at , then is differentiable at .
First, we observe that and are polynomials and so are differentiable everywhere.
Now you noted that for all . However, You also observed that only along the coordinate axes (i.e. when or when ). So the Cauchy-Riemann Equations are only satisfied along the coordinate axes. By our above Theorem, the function is differentiable at points along the coordinate axis.
Now here's the key: We only have that is differentiable at points along the axes, we do NOT have that is differentiable on neighborhoods/little disks of such points. Since we can't find entire disks on which the function is differentiable that means the function is not holomorphic (by definition of holomorphic - see above) and so is not analytic.
For example, the point sits on the imaginary axis, because there. Since sits on the coordinate axis, is differentiable at this point (by our above argument). However, if we draw a little disk around (NO MATTER HOW SMALL) we will include points that do not sit on the coordinate axis. Since the points that don't sit on the coordinate axis fail to satisfy the Cauchy-Riemann Equations (specifically such points WON'T satisfy ), is not differentiable at them.
To sum up, the important feature that makes the given function NOT analytic/holomorphic is the fact that we can't say is differentiable on ENTIRE neighborhoods surrounding points, all we can say is that is differentiable at specific points.
Does this help clear things up?