Dirichlet's test for convergence of improper integrals

Solution 1:

I paste here a theorem from Mathematical Analysis by S. C. Malik,Savita Aror (see also Google Books here)

Theorem. If $\phi$ is bounded and monotonic in $[a,+\infty)$ and tends to zero at $+\infty$, and $\int_a^X f$ is bounded for $X \geq a$, then $\int_a^\infty f \phi$ is convergent.

As you can see, no continuity is really necessary. Honestly, the continuity assumption appears often for the sake of simplicity. Improper integrals can be defined as limits of Riemann integrals: all you need is local integrability. However, we know that continuity is "almost necessary" to integrate in the sense of Riemann, so teachers do not worry too much about the minimal assumptions under which the theory can be taught.