When you simplify the rational expression, one of the factors divides out. When that happens, it is not an asymptote.
Point of Discontinuity
Definition of Point of Discontinuity
A function is said to have a point of discontinuity at x = a or the graph of the function has a hole at x = a, if the original function is undefined for x = a, whereas the related rational expression of the function in simplest form is defined for x = a.
The above function in its original state is not defined for x=0 nor x=1.
In the simplified form above, the function is still not defined for x = 0.
Therefore, x = 0 is an asymptote. However, it is defined for x = 1.
Plugging x=1 into the simplified function, we find f(1)=-3.
Thus we can say that the function f(x) has a point of discontinuity at (1, -3).