Why will $u$ and $v$ have continuous partial derivatives of all orders if $f$ is an analytic function in Ahlfors' Complex Analysis?

Question

In Ahlfors' Complex Analysis, I have learned the fact that if $f(z)=u(x,y)+iv(x,y)$ is an analytic function, then $f^\prime(z)$ exists and all the partial derivatives of $u$ and $v$ exist.

Then Ahlfor assumes that we have proved that the derivative of an analytic function is itself analytic. By using this result, it is easy to know that $u$ and $v$ will have partial derivatives of all orders. But how to understand that these partial derivatives (of all orders) are continuous?