if $\int\limits_a^bf(x)dx=0$ for all rational numbers $a<b$, then $f(x)=0$ a.e
88690
$\int_a^b f(x)\mathrm dx=0$ for all $-\infty<a<b<\infty$ a,b rational implies $f=0$ a.e
80710
Does $\int_{-1}^1 \frac 1 x dx$ equal zero?
136966
Riemann integral proof $\int^b_a f(x) \, dx>0$
112323
Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$
50472
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
21954
A question about $f(x)\equiv 0$
34187
The limit of the derivative of an increasing and bounded function is always $0$?
81
$\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$
104394
What does $\int_{a}^{b} f(x)$ mean?
121663
If $a$ is a zero divisor of $\mathbb{F}$, then does $a$ fail to have a multiplicative inverse?
33081
If $\int_{\Bbb R^2}|f(x+iy)|\,dx\,dy<\infty$ then prove that $f\equiv 0$
95260
Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.
69008
$f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
105569
If $f$ is integrable in $[a-1,b+1]$, then prove that $\lim \limits_{h\to 0} \ \int_{a}^{b}|f(x+h)-f(x)| ~ dx=0$
5180
How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
51594
$\lim_{n→∞} f(n+x)=0$ results in $ \lim_{x→∞} f(x)=0$
8854
Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$