If $(f_n):[0, 1] \to [0, 1]$ are continuous and converge to $f$ pointwise, must $f$ be Riemann Integrable?
1014
Pointwise limit of continuous functions not Riemann integrable
111211
Does $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}=1$ imply that $a_n$ convergent?
6990
For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
41057
Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?
29180
If $f(x/n)\to0$ when $n\to\infty$, for every $x$ and $f$ is continuous, then $f(x)\to0$ when $x\to0$, or not?
111990
Continuous function from $(0, 1]$ onto $(0, 1)$?
33050
proof that translation of a function converges to function in $L^1$
23942
If a function $f(x)$ is Riemann integrable on $[a,b]$, is $f(x)$ bounded on $[a,b]$?
146921
If |f| is Riemann integrable, then f is Riemann integrable???
138397
If a sequence of functions $f_n$ converges uniformly to $f$, does this imply $f_n$ is continuous? How to prove?
1332
Pointwise convergence and convergence of integrals implies $L^1$ convergence
148410
How do I show $f_n(x)=n^2 x^n(1-x)$ pointwise converges to $0$ on $[0,1]$?
155822
Does uniform convergence preserves absolute continuity?
99485
Prove that $1/f$ is Riemann integrable on $[a,b]$
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
110415
Proving a sequence of step functions converges pointwise to a function $f$.
151936
Pointwise Convergence and Convergence in the mean of Riemann-integrable functions