I will cite the relevant definitions and theorems I am working with from Wunsch's Complex variables with applications.
Def. (Analyticity) A function $f(z)$ is analytic at $z_0$ if $f'(z_0)$ exists not only at $z_0$ but at every point in a neighbourhood of $z_0$
Def. (Analyticity in a Domain) If a function is analytic at every point belonging to some domain we say that the function is analytic in that domain.
Corollary 1 (From C-R Equations) : If $f$ is differentiable at $z_0$, $f'(z_0) = \frac{\partial{u}}{\partial{x}} + i\frac{\partial{v}}{\partial{x}} = \frac{\partial{v}}{\partial{y}} - i\frac{\partial{u}}{\partial{y}}$
Theorem 1 : Let $f(z) = u(x,y) + i v(x,y)$ where $z=x+iy$. If $u,v$ and their first partial derivatives $(\frac{\partial{u}}{\partial{x}}, \frac{\partial{v}}{\partial{x}}, \frac{\partial{u}}{\partial{y}}, \frac{\partial{v}}{\partial{y}})$ are continuous throughout some neighbourhood of $z_0$, then the satisfaction of the Cauchy-Riemann equations at $z_0$ is both sufficient and necessary for the existence of $f'(z_0)$.
Theorem 2(Cauchy-Goursat) If $f$ is an analytic function in a region $R$ containing a simple closed contour $\mathcal{C}$ and its interior. Then, $$\int_{\mathcal{C}} f(z) dz = 0$$
I was reading the proof for the Cauchy-Goursat theorem and at first the author proves the theorem for a special case (Using Green's Theorem of real variables) in which we assume $f'(z)$ is continuous in $R$.
This is where my confusion arises. If $f$ is analytic in the whole region $R$ doesn't Theorem 1 imply that it has first partial derivatives continuous in $R$? And if $f$ has continuous partial derivatives, from Corollary 1, $f'$ is the sum of continuous functions at every point in $R$ so $f'$ is continuous too.
So my question is, why is this even a special case, doesn't analyticity in a domain imply the first derivative is continuous? Where am I going wrong in this?
Update: I think my error was in interpreting Theorem 1. At first I read it as,
Partial derivatives are continuous and C-R equations are satisfied $\iff$ $f'(z_0)$ exists.
I now see this is not what it was saying. The correct reformulation would be,
Consider $F : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ s.t $F(x,y)= (u(x,y),v(x,y))$.Then,
$F$ is real differentiable and C-R equations are satisfied $\iff f$ is complex differentiable
Can someone verify this was mistake and the what I stated just now is correct?
No, Theorem 1 says: Assume that the first partial derivatives of $u,v$ exist and are continuous in a neighborhood of $z_0.$ Then $f'(z_0)$ exists iff $u,v$ satisfy the Cauchy Riemann equations at $z_0.$