Undergrad Find potential integrating on segments parallel to axes

[crick]

A simple method to find the potential of a conservative vector field defined on a domain $D$ is to calculate the integral
$$U(x,y,z)=\int_{\gamma} F \cdot ds$$

On a curve $\gamma$ that is made of segments parallel to the coordinate axes, that start from a chosen point $(x_0,y_0,z_0)$.

I would like to know what are precisely the conditions that $D$ must satisfy to use this method. $D$ should be made in such way that "any point can be connected to $(x_0,y_0,z_0)$ with, indeed, a segment parallel to the coordinate axes".

But what are the sufficient mathematical conditions for $D$ in order to have this property?**

I would say that it surely has to be connected, but that seems not to be enough. For example taking
$$D= \{ (x,y) : y>x-1\} \,\,\,\, \,\,\,\,\,(x_0,y_0)=(0,0)$$
$D$ is connected but I do not think that any point can be connected to $(0,0)$ via a segment parallel to the coordinate axes.

 

[haruspex]

I do not think that any point can be connected to (0,0) via a segment parallel to the coordinate axes.

It does not need to be. It only has to be connected by some path consisting of segments parallel to the axes. Of course, it could get quite messy.
A simple example which does not work is two regions connected only by a diagonal line.

 

[Svein]

In my mind it seems like a compactness argument should be useful, but I do not presently have the energy to follow up.