A vertical asymptote can be, but need not be a discontinuity. In
simple terms, the distinction depends whether the domain extends on
only one side of the (no discontinuity) or both sides (infinite
discontinuity).
For example, there is no discontinuity in f(x) = 1/x for x >
0
On the other hand, f(x) = 1/x for x ≠0 has an infinite
discontinuity at x = 0.