Counter-examples to the Integral Limit Theorem if you only assume $f_n \rightarrow f$ point wise. This is the theorem I have.
Integral Limit Theorem
Assume that $f_n \rightarrow f$ uniformly on $[a,b]$ and that each $f_n$ is integrable. Then, $f$ is integrable and
$\lim_{n\to\infty} \int_{a}^{b}f_n = \int_{a}^{b}f$
What are some counter-examples to the Integral Limit Theorem if you only assume $f_n \rightarrow f$ pointwise? I honestly do not have any possible solutions.
Thank you!
 A: Here are some choices for what is actually a big topic and attracted much attention during the 19th century.
Let us restrict to just the Riemann integral on an interval  $[a,b]$.  We are skipping over examples for the improper Riemann integral (unbounded functions).  We dare not enter into the world of the Lebesgue integral where this problem truly belongs (not yet anyway).
So I won't give you any examples.  But I will tell what examples we want.
A.   If  $\{f_n\}$ is a sequence of continuous functions that converges pointwise to a function $f$ it is possible that  $f$ is unbounded.
B.   If  $\{f_n\}$ is a sequence of continuous  functions that converges pointwise to a bounded function $f$ it is possible that, even though it is bounded,  $f$ is not Riemann integrable.
C.   If  $\{f_n\}$ is a sequence of continuous functions that converges pointwise to a Riemann integrable function $f$ it is possible that
$$\lim_{n\to\infty}  \int_a^bf_n(t)\,dt $$
does not exist.
D.   If  $\{f_n\}$ is a sequence of continuous  functions that converges pointwise to a continuous  function $f$ it is possible that
$$\lim_{n\to\infty}  \int_a^bf_n(t)\,dt $$
does exist but does not equal $\int_a^b f(t)\,dt$.
Contrast this with a famous theorem of the Italian mathematician Cesare Arzelà from 1885.

Theorem.    If  $\{f_n\}$ is a uniformly bounded sequence of Riemann integrable functions on an interval $[a,b]$ that converges pointwise on $[a,b]$ to a Riemann
integrable function $f$ then $$\lim_{n\to\infty} 
 \int_a^bf_n(t)\,dt=\int_a^b f(t)\,dt.$$

[See https://sites.math.washington.edu/~morrow/335_16/dominated.pdf for a discussion.]
