Show that a function is defined on all of $\mathbb{C}$ but nowhere analytic.
$$z \mapsto2xy+i(x^2-y^2)$$ How would I go about and show that a function is defined. We are currently learning Cauchy-Riemann's, so I am assuming that I could probably show that it is not analytic by Cauchy-Riemann, but I am not sure of how to show a function is defined.
General rule: to show a function is defined, you need to show that under the indicated rule of mapping, each element in the domain can get mapped to one and only one element in the codomain. In your case this is obvious. Some not so obvious cases include functions involving quotient maps etc. To show non analyticality, yes you are right about using Cauchy Riemann criterion.