# Switching the limit and integral Criteria

Does there exist a sequence of function $$f_n$$ on $$[a, b]$$ such that $$f_n$$ is uniformly convergent, $$f_n$$ is not continuous, and

$$\lim_{n \to \infty} \int_a^b f_n(x) \mathop{dx} \neq \int_a^b \lim_{n \to \infty} f_n(x) \mathop{dx}?$$

In particular, can we switch the integral and limit whenever $$f_n$$ is uniformly convergent or do we need $$f_n$$ to also be continuous? If there does exist such a sequence of functions, please provide an example.

• I do not believe continuity of the $f_n$ is required, just (Riemann) integrability on $[a,b]$. One must show that $f=\lim_{n\to\infty}f_n$ is integrable, then the same proof shows that we may interchange the limit with the integral. – Math1000 Nov 21 '19 at 22:23

## 2 Answers

No because of the following theorem:

If $$(f_{n})_{n=1}^{\infty }$$ is a sequence of Riemann integrable functions defined on a compact interval $$I$$ which uniformly converge with limit $$f$$, then $$f$$ is Riemann integrable and its integral can be computed as the limit of the integrals of the $$f_{n}$$: $$\int _{I}f=\lim _{n\to \infty }\int _{I}f_{n}.$$

There is no continuity assumption in the theorem here.

I assume you talk about riemann integrals. The $$f_n$$ don't need to be continous. But you need that they are riemann integrable. See wikipedia.