# Continuity of function that is expressed as the subtraction of two integrals

Let $$\rho\in L^1(R)$$ be a given function. We associate it with $$U(x) = \frac{1}{2}\int_x^{+\infty}\rho(y)dy - \frac{1}{2}\int_{-\infty}^x\rho(y)dy.$$

Show that $$x \mapsto U(x)$$ is a continuous function.

My ideas:

Calculate $$|U(x)-U(y)|$$ and find an expression of type $$C| x-y |$$ using the mean value theorem for integrals.

• It suffices to show "one of the integrals is continuous". You can use the uniform integrability of $|\rho|$ to do that. Dec 15 '20 at 14:31
• See this. Dec 15 '20 at 14:47
• I go to see the post that you give me Dec 15 '20 at 14:50