I am dealing with the following function:
$$m(t) = \min_{x \geq 0} e^{t x} \cdot \int_0^\infty e^{-x u} f(u) du,$$
where $t \geq 0$ and $f(u)$ is a pdf of a random variable $X$ distributed over $[0, \infty)$. It is clear that $m(0) = 0$, and $m(t)$ is increasing in $t$.
My question is: Is the function $m(t)$ continuous at $t = 0$? I conjecture that this can be bounded by some linear function, i.e., $m(t) \leq c \cdot t$ for constant $c > 0$, but I am unsure how to do.
Thank you in advance.