I am working through critical points and finding extrema. I understand that if a point is not in the domain of a function, then it is not a critical point of that function and by definition therefore cannot be an extremum or inflection point of that function — but can a function change signs either side of a point at which it is undefined?

When I test for the sign of the derivative, should I include not only the intervals between the critical points but also either side of points where f is undefined? Is there a non-piecemeal function that increases to the left of a vertical asymptote and decreases to the right of it?