If g(a) is discontinuous at x=a and f(a) is continuous at x=a, what can we say about continuity of g(x)f(x) at x=a?

Many of the examples I have seen are about continuity are over an entire domain instead of a specific point and the most common counterexample I have seen is to set f(x)=0 over the domain, then f(x)g(x) will always be continuous since it is just the constant function 0.

However, if we are only talking about continuity at a specific point, f(a)= 0 would not be enough for us to conclude that f(a)g(a) is continuous at x=a since any points in the neighborhood of a can be mapped to values other than 0 by f(x). Is this correct?