CR equations and differentiability

[strangequark]

Homework Statement

Where is f(z) differentiable? Analytic?
$$f(z) = x^{2} + i y^{2}$$

Homework Equations

Cauchy-Riemann Equations

The Attempt at a Solution

I calculated the partial derivatives,

$$u_{x} = 2x$$
$$v_{y} = 2y$$
$$u_{y} = 0$$
$$v_{x} = 0$$

Then said that for the CR equations to hold,

$$u_{x}=v_{y}$$ therefore $$y=x$$
and
$$u_{y}=-v_{x}$$ therefore $$0=0$$

Then because the partial derivatives are continuous for all $$x,y$$, $$f(z)$$ is differentiable along $$y=x$$

$$f(z)$$ is nowhere analytic because an arbitrarily small open disk centered at any point on the line $$y=x$$ will always contain points which are not differentiable.

Is that sufficient to show differentiability? Or am I misapplying the cauchy-riemann conditions?

 

[strangequark]

I suppose what I'm really asking is wether or not I need to look at the limits depending on the direction of approach, or if this is sufficient as is? Help?

 

[Dick]

I think that is good enough. You could explicitly show the derivative limit is independent of direction along x=y, but why? That's what CR are for.