Q:

What's the meaning of "twice continuously differentiable"?

The picture is from Complex Analysis by Stein.
What's the meaning of "twice continuously differentiable"?

A:

It means that the functions $\dfrac{\partial^2u}{\partial x^2}$, $\dfrac{\partial^2u}{\partial x\partial y}$, and $\dfrac{\partial^2u}{\partial y^2}$ exist and are continuous.

A:

Twice continuously differentiable means the second derivative exists and is continuous. 

