Is my methodology for checking differentiability and analyticity correct?

[NewtonianAlch]

Homework Statement

State the Cauchy-Riemann equations and use them to show that the function defined by f(z) = |z|^2 is differentiable only at z = 0. Find f′(0). Where is f analytic?

The Attempt at a Solution

f(z) = |z|^2 = (x^2 + y^2)

$\frac{du}{dx}$ = 2x, $\frac{dv}{dy}$ = 0

$\frac{du}{dy}$ = 2y, $\frac{dv}{dx}$ = 0

So x and y must equal 0 for the C.R equations to hold, thus z = 0 + 0 = 0

So this proves they are only differentiable when z = 0.

f'(0) = 2|0| = 0

I'm not sure about the analytic part, f(z) = |z|^2 would be a parabola, so if it's only differentiable when z = 0, i.e. at the origin then it can't be analytic, because an e-neighbourhood at that point will yield points not on the parabolic line, therefore it's not analytic. Is this right or wrong?

 

[NewtonianAlch]

I just found out that f'(z) = $\frac{du}{dx}$ + i$\frac{dv}{dx}$

So according to this, f'(z) = 2x + 0 = 2x

Which is 2(Re(z)), not the same thing as what I said earlier about f'(z) = 2|z| = 2$\sqrt{x^2 + y^2}$

In either case f'(0) = 0

 

[NewtonianAlch]

Hmm, just went over it again and realized, that if it's differentiable only when x = 0, and y = 0, then it's only differentiable when it's on the axes, and hence not analytic because an e-neighbourhood of any size on the axis is a point outside the axis, and hence not differentiable.