What's the difference between analytic and continuously differentiable?

[AxiomOfChoice]

What's the difference between "analytic" and "continuously differentiable?"

I'm reading Gamelin's Complex Analysis book, and he talks about $f(z)$ being analytic if it is continuously differentiable and satisfies the Cauchy-Riemann equations. But if $f(z)$ is continuously differentiable, doesn't that mean $f'(z)$ 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 $f(x,y) = (u(x,y),v(x,y))$ exist and are continuous. This, of course, does NOT imply complex differentiability (just consider $f(z) = \overline z$); we also have to have the CREs satisfied. But complex differentiability DOES imply real differentiability...am I right?

 

[HallsofIvy]

I'm reading Gamelin's Complex Analysis book, and he talks about $f(z)$ being analytic if it is continuously differentiable and satisfies the Cauchy-Riemann equations. But if $f(z)$ is continuously differentiable, doesn't that mean $f'(z)$ 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 $f(x,y) = (u(x,y),v(x,y))$ exist and are continuous. This, of course, does NOT imply complex differentiability (just consider $f(z) = \overline z$); we also have to have the CREs satisfied. But complex differentiability DOES imply real differentiability...am I right?

I'm not sure what you mean by "complex differentiability" and "real differentiability". With "real differentiability" are you thinking of the derivatives of the real and imaginary parts of f(z) with respect to the real and imaginary parts of z? If that is what you mean, then yes, the existence of df/dz implies the existence of the partial derivatives of the real and imaginary parts.

However, "f'(z) exists and is continuous" is NOT, in my opinion, "the very definition of analyticity"- though it can be used as a definition. A function is analytic at a point if its Taylor series exists at that point and is equal to the function on some neighborhood of that point. Obviously, if the Taylor series exists, all derivatives exist and so are continuous. It is not so obvious, but can be proved, that if the first derivative exists, then all must exist.