Show that $f$ is identically zero if and only if $\int_a^b f(x)dx = 0$
69695
$f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$
48314
Show that if a function is not negative and its integral is $0$ than the function is $0$
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
145573
How to show that $f$ is a zero function?
33081
If $\int_{\Bbb R^2}|f(x+iy)|\,dx\,dy<\infty$ then prove that $f\equiv 0$
31522
Proving $f$ has at least one zero inside unit disk
100765
If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
88690
$\int_a^b f(x)\mathrm dx=0$ for all $-\infty<a<b<\infty$ a,b rational implies $f=0$ a.e
18261
Prove $\int_{a}^{b}{f(x)dx}>0$.
40047
Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$?
82588
If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere
150502
If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$
16109
Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
39921
show that f is constant on I
71225
$f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
112323
Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$
131793
Zero integral of measurable $f$ on every interval implies $f=0$?