Let $f : [0, 1] \to \mathbb{R}$ be a monotonic function such that the number of its discontinuity points is finite. For $t \geq 0$ let $m_t = \int_0^1 x^t f(x)dx.$ Then which of the following statements is true?
(A) There exists a $t \geq 0$ for which $m_t$ is not defined.
(B) $m_t < \infty$ for all $t \geq 0.$
(C) $m_t$ is defined and is equal to $\infty$ for all $t \geq 0.$
(D) $m_t$ is defined for all $t \geq 0$ and there exists $t_1,t_2 \geq 0$ such that $m_{t_1} = \infty$ and $m_{t_2} < \infty.$
I was thinking of $m_t$ like the $t^{th}$ order moment of a random variable $X$ but I'm not sure of the range and domain, and the question is about existence of the moments. Can someone help me proceed?