Neighborhood of an analytic function

Question

I have two conceptual questions I'd like some clarification on.

1. Suppose we have a complex function and its partial derivatives are continuous and the Cauchy Remain equations are satisfied iff x=y. Is this function considered Analytic? How can we define its neighborhood? Should the neighborhood be 2 dimensional (not a line) thus implying not analytical?

2. Suppose we have a complex function and its partial derivatives are not continuous at x=0 (suppose it makes the denominator zero), but the Cauchy Reiman equations are satisfied everywhere. Is this function considered analytic?