Is the lebesgue integral of a measurable function continuous?

Question

I was wondering if the lebesgue integral of a measurable function at least continuous? What kind of regularity on the integrand do we need for it to be absolutely continuous so that we can say its differential almost everywhere?

Answer 1

The integral of an integrable function is a single number, so asking whether it's continuous makes little sense. Presumably you're asking about the continuity of $$F(x)=\int_{-\infty}^x f(t)\,dt$$or maybe $\int_0^x$.

If $f$ is integrable then $F$ is absolutely continuous; this is one of the main theorems in this context. If $f$ is just locally integrable then it follows that $F(x)=\int_0^x f$ defines a locally absolutely continuous function.