There was a similar question posted here but I'm a bit confused by what "nowhere analytic" means.
In this question, we're given $f(x)=x^2+iy^2$. The first part of the question is to show that $f(x)$ is differentiable at all points on the line $y=x$, which I did by showing that $f_x$ and $f_y$ are both continuous and that $f_y=if_x$ when $y=x$.
For the next part, we had to show that $f(x)$ is nowhere analytic. From the Cauchy-Riemann equations, a function is analytic only if $u_x=v_y$ and $u_y=-v_x$. Taking the partials, I got:
$$u_y=-v_x\rightarrow0=0$$
$$u_x=v_y\rightarrow2x=2y$$
Wouldn't this mean that the function satisfies the Cauchy-Riemann equation whenever $y=x$ and is thus analytic along that line? In that case, how is $f(x)$ nowhere analytic?
Best Answer
So, does the set $x=y$ contain any open sets in $\Bbb C$?