Cauchy-Riemann equations and required continuity of derivatives So I just read, that for any analytic function, the Cauchy-Riemann equations will hold. However, the reverse, i.e. Cauchy-Riemann equations hold -> function is analytic, is supposedly only true if the partial derivatives are continuous.
What would be an example though? I cannot think of any function, where the Cauchy-Riemann equations would hold, while not being analytic.
Can someone please help me out?

Take any analytic function defined on $\mathbb C$. Now remove all the parts except those which are on the coordinate axes, so only a kind of cross remains. Replace the removed parts by some horribly discontinuous mess, such that the new function is also discontinuous at $0$. The function obtained this way will still be partially differentiable at $0$, and the partial derivatives will still be the same as those of the original function, since the partial derivatives at $0$ only depend on how the function behaves on the coordinate axis, which we didn't change.
A concrete example:
$$f(z)=\begin{cases}0&z\textrm{ is on the coordinate axes}\\0&z\textrm{ has both rational imaginary and real part}\\1&\textrm{otherwise}\end{cases}$$
