Cauchy-Riemann Equations and differentiability at a single point I am trying to figure out when can we conclude a complex function $f$ is differentiable at a point (and only that point). For example the function $|z|^2$ is differentiable only at $z=0$. However, I am not sure if I can use CR. The version we went over in class states that $f$ is differentiable on an open set $O$ if for each $z \in O$ we know C-R are satisfied and that the derivatives are continuous on $O$. However, a single point is clearly not an open set, so I do not know if C-R are satisfied at $z_0$, and $u,v$ are continuously differentiable at $z_0$, then f is differentiable at $z_0$.
$f(z)=|z|^2=x^2+y^2$ doesn't satisfy the C-R equations in a nbhd of $0$.
Thus you need a different way to conclude that $f$ is differentiable there.

See "Complex Differentiable -- from Wolfram MathWorld" http://mathworld.wolfram.com/ComplexDifferentiable.html
$f(z)=|z|^2=x^2+y^2$ doesn't satisfy the C-R equations in a nbhd of $0$.
Thus you need a different way to conclude that $f$ is differentiable there.

See "Complex Differentiable -- from Wolfram MathWorld" http://mathworld.wolfram.com/ComplexDifferentiable.html