Can you define what you mean here by discontinuity? I meant that the function of the gravitational field is continuous everywhere under the topology of the mani

Can you define what you mean here by discontinuity? I meant that the function of the gravitational field is continuous everywhere under the topology of the manifold, the mathematical definition. I see no discontinuity, but perhaps you're using a different definition? The "no gravity" line there means the stress tensor is zero, and this is a so-called "vacuum solution." Essentially all closed form solutions to the field equations are vacuum solutions because it's almost impossible to write a closed form solution otherwise.