You have

> f'(x)/f(x) = 1/g(x)

which implies

> log(f(x)) = H(x) := ∫dx/g(x)

Here, H(x) should exist on some appropriate domain which excludes any roots of g. As g is continuous, so will it's reciprocal be, and hence an antiderivative would exist. An explicit solution would be

> f(x) = C\*e^H(x)

where C comes from the constant of integration.