Yes, the biggest giveaway to understanding this problem is what can does a fraction have restrictions on. Since x cannot be 0, in the example: 1/x, then, the domain is all real numbers, except for 0.
So, to tackle this problem, let's consider the restrictions of g(x). Now, we know that g(x) cannot be zero, since that will render the fraction indeterminate. So, we know g(x) ≠ 0. But that's not the same as g(0).
If there are no restrictions on g(x), then the function will be continuous for all x.