Converse of the Cauchy-Riemann condition

Question

The Cauchy-Riemann condition states that an analytic function satisfies: \begin{split} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y}; \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x} \end{split} The converse of the statement requires additional conditions: the first partial derivatives of $u, v$ exist and are continuous.

My question is: When a function satisfies the Cauchy-Riemann condition, shouldn't it have already satisfied the above additional conditions? Otherwise, it wouldn't satisfy the Cauchy-Riemann condition if its partial derivatives do not exist. Am I missing something fundamental here?

Answer 1

Let $f(x+iy)=\sqrt {|xy|}$. You can verify that the partial derivatives all exist at $0$ and satisfy the C-R equations but f is not differentiable at 0.

Answer 2

The converse of the Cauchy-Riemman conditions require that $u,v$ be differentiable. This implies in particular that the partial derivatives exist, but does not require its continuity.