$f(z)=f(x+iy)=u(x,y)+iv(x,y)$
$f(z)=f(x+iy)=f(re^{i\theta})=f(r\cos\theta+ir\sin\theta)=u(r\cos\theta,r\sin\theta)+iv(r\cos\theta,r\sin\theta)=\varphi(r,\theta)+i\psi(r,\theta)$
$\varphi(r,\theta)=u(r\cos\theta,r\sin\theta)$
$\psi(r,\theta)=v(r\cos\theta,r\sin\theta)$
$r\frac{\partial\varphi}{\partial r}=\frac{\partial\psi}{\partial\theta}$
$\frac{\partial\varphi}{\partial\theta}=-r\frac{\partial\psi}{\partial r}$
(The two partial differential equations above are called Cauchy-Riemann equations for $r$ and $\theta$.)
$\frac{\partial\varphi}{\partial r}, \frac{\partial\varphi}{\partial\theta}, \frac{\partial\psi}{\partial r}, \frac{\partial\psi}{\partial\theta}$ are continuous.
Assume the conditions above are all satisfied. We want to prove $f$ is analytic.
Proof:
I'm confused about how to prove rigorously that all the first-order partial derivatives of $u,v$ are existed and continuous.
But if all the first-order partial derivatives of $u,v$ are existed and continuous, we have:
$\frac{\partial\varphi}{\partial r}=\frac{\partial u}{\partial x}\cos\theta+\frac{\partial u}{\partial y}\sin\theta$
$\frac{\partial\varphi}{\partial\theta}=-\frac{\partial u}{\partial x}r\sin\theta+\frac{\partial u}{\partial y}r\cos\theta$
$\frac{\partial\psi}{\partial r}=\frac{\partial v}{\partial x}\cos\theta+\frac{\partial v}{\partial y}\sin\theta$
$\frac{\partial\psi}{\partial\theta}=-\frac{\partial v}{\partial x}r\sin\theta+\frac{\partial v}{\partial y}r\cos\theta$
Then, we have:
$\frac{\partial u}{\partial x}\cos\theta+\frac{\partial u}{\partial y}\sin\theta=-\frac{\partial v}{\partial x}\sin\theta+\frac{\partial v}{\partial y}\cos\theta$
$-\frac{\partial u}{\partial x}\sin\theta+\frac{\partial u}{\partial y}\cos\theta=-\frac{\partial v}{\partial x}\cos\theta-\frac{\partial v}{\partial y}\sin\theta$
Let $\theta=0$, we get Cauchy-Riemann equations for $x$ and $y$:
$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
Since $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y} $are also continous (Assume we only have learnt first three chapters of Ahlfors' Complex Analysis. So we need these continuities), We conclude that $f$ is analytic.
The key point is we should prove all the first-order partial derivatives of $u,v$ are existed and continuous. But I don't know how to write a rigorous proof for this.