I was going through the introductory chapter on PDE, and the following highlighted text does not make any sense to me:
$$F\left(x,y,z,\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right)=0\tag 1$$ A solution of Eq. (1) in some domain $\Omega$ of $\mathbb{R} ^2$ is a function $z = f(x, y)$ defined and is of C' in $\Omega$ should satisfy the following two conditions:
In my current capacity I think that this highlighted text is incomplete, and hence not making sense. If not please tell me what does it mean? If incomplete please let me know what the author was trying to convey there.
The image of the whole page is attached below, for further reference.
