If a function is continuous everywhere, but undefined at one point, is it still continuous?

$G$ is continuous on the domain $[0,3)\cup(3,6]$.

Referring to the aforementioned definition (1) that the limits converge to the actual value at this point.

3 is not in the domain. For every point in the domain of $g$, we have the required convergence.

The function is continuous everywhere in the interval except that point deleted from the domain, it's more a nuance of the language than anything else. Choose any point that is not $3$ in that interval: you can then find left- and right-hand limits to that point and show they're equal.