Is continuity required for the comparison test of improper integrals?

In my lecture notes, the comparison test for improper integrals is given in the following way:

"Suppose that f and g are continuous functions with f(x) >= g(x) >= 0 for all x >= a. Then if the improper integral of f(x) from a to infinity is convergent, then the improper integral of g(x) from a to infinity is convergent"

However, I just came across these notes that do not mention continuity as a requirement to apply the comparison test (check out theorem 18.3 on the end of page#2), just non-negativity of f(x) and g(x)

So, can i use the comparison test if i don't know whether f(x) and g(x) are continuous?