How can I prove the continuity or discontinuity of the following $\mu$ as a function of $x$?
$$\mu(x) = \int_0^x\exp\left\{ -\int_0^t\frac{d\alpha(s)}{\beta(s)+1} \right\}dt + \int_x^{+\infty}\exp\left\{ -\left(\int_0^x\frac{d\alpha(s)}{\beta(s)+1} + \int_x^{t}\frac{d\alpha(s)}{\beta(s)} \right)\right\}dt$$
$\alpha(\cdot)$ is a finite positive measure and $\beta(\cdot)$ is a finite positive function. I do not need to solve the integral.
Thanks for the answers.
