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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.

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  • $\begingroup$ 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. $\endgroup$ – Math1000 Nov 21 '19 at 22:23
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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.

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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.

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