# Blog

## What are the non-integrable functions of rational numbers?

• These are non-integrable. Functions that have discontinuities of positive measure can not be Riemann integrated, for instance the characteristic function of the rational numbers on the unit interval. However, a more general notion of integral, the Lebesque integral, includes this function and many other that can not be Riemann integrated.

## Can funfunctions be Riemann integrated?

• Functions that have discontinuities of positive measure can not be Riemann integrated, for instance the characteristic function of the rational numbers on the unit interval. However, a more general notion of integral, the Lebesque integral, includes this function and many other that can not be Riemann integrated.

## Are all integrable functions analytic functions?

• Since analytic functions are very "nice" and the only requirement for integrability is that the function be bounded and have discontinuities only on a set of measure 0, jumping form integrable to analytic leaves out "almost all" integrable functions! What function are you talking about? And what do you mean by "expanding it over infinite terms"?