Dirichlet's test claims that for two continuous functions $f,g\in[a,\infty]$ where $f,g\geq 0$, if a certain $M$ exists such that $\left|\int_a^bf(x)dx\right|\leq M$ for every $a\leq b$, and $g(x)$ is monotonically decreasing, and $\lim_{X\to\infty}g(x)=0$, then $\int_a^\infty fg$ is convergent.
Does this also apply for a non-continuous $f(x)$? $g(x)$ is still continuous.
This question relates to another question of mine, regarding a specific integral problem. If this is true, then my other problem will be solved.