I know that if the partial derivative of real and imaginary parts of a complex function satisfy the cauchy-riemann equations and the partial derivatives are continuous at a point then the function analytic at the point, but I have an example which contradicts to this if I don't have a mistake;
Assume $f(z)=1$ if $z$ is on the $x$-axis and $y$-axis $0$ otherwise. The limit $\lim \frac{f(z)-f(0)}{z}$ doesn't exist where $z\rightarrow 0$ so it is not analytic at $0$ but the partials are continuous and satisfy Cauchy-Riemann equations at $0$. There should be some mistake but I couldn't able to see it.
Best Answer
Don't you need continuity of $f$? As far as I know the theorem is as follows: A continuous function $f$ is analytic if the partial derivatives are continuous and satisfy the Cauchy-Riemann equations.