Prove that $f(x)=\begin{cases}0, & x\notin \mathbb{Q}\\ \frac{1}{q}, & x=\frac{p}{q} \text{ in lowest terms}\\ \end{cases}$ is integrable on $[0,1]$.
5814
Integrability of Thomae's Function on $[0,1]$.
6825
If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.
155023
Is $\frac{1}x\cos\left(\frac{1}x\right)$ Lebesgue integrable on $[0,1]$?
6990
For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
94869
Show that $ \int_{k}^{2} \frac{1}{f(y)} =1$
157715
Proving $f$ is Lebesgue integrable iff $|f|$ is Lebesgue integrable.
124061
Prove the reciprocal of a function is integrable if it is bounded
68811
If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.
95469
Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
70304
Prove there are $\xi$, $\eta$ with $f'(\xi)f'(\eta)=1$
46310
Showing that $ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dx=\ln2 $
41173
If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
58647
Show that $f$ is integrable and $\int_{I} f=0$
31319
An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.
105569
If $f$ is integrable in $[a-1,b+1]$, then prove that $\lim \limits_{h\to 0} \ \int_{a}^{b}|f(x+h)-f(x)| ~ dx=0$
77106
If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ ...
91775
Show that $\int_0^1 f^2(x)dx\geq 3$, if $f:[0,1]\to \mathbb{R}$ is an integrable function s.t. $\int_0^1 f(x)dx=\int_0^1 xf(x)dx=1$