The relationship between Cauchy-Riemann equations and differentiability

Question

What is the relationship between Cauchy-Riemann equations and differentiability of a complex valued function at any point $z=z_0$? If Cauchy-Riemann equations are satisfied at $z=z_0$ then can we say that $f(z)$ is differentiable at $z=z_0$?

Answer 1

The reply is probably too late, but:

According to the theorem, that f(z) satisfies Cauchy-Riemann equation is the necessary condition for the complex function to be complex differentiable. The sufficient conditions also require all first order partial derivatives of u, v (where f(x,y)=u(x,y) + iv(x,y)) are continuous (at z=z_0.)

Hence, complex differentiability implies f satisfying C-R equation, but not the other way around. Here I found an counter example: Show the Cauchy-Riemann equations hold but f is not differentiable.