What's the difference between "analytic" and "continuously differentiable?" I'm reading Gamelin's Complex Analysis book, and he talks about [itex]f(z)[/itex] being analytic if it is continuously differentiable and satisfies the Cauchy-Riemann equations. But if [itex]f(z)[/itex] is continuously differentiable, doesn't that mean [itex]f'(z)[/itex] exists and is continuous, which is the very definition of analyticity? There's obviously something I'm missing here. I think it's that "continuously differentiable" means that the partials of [itex]f(x,y) = (u(x,y),v(x,y))[/itex] exist and are continuous. This, of course, does NOT imply complex differentiability (just consider [itex]f(z) = \overline z[/itex]); we also have to have the CREs satisfied. But complex differentiability DOES imply real differentiability...am I right?