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 becuase 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.