$f = 0$ a.e. in $[0, T]$ if $\int_0^t f(s) ds = 0$ for all $t \in [0, T]$?
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.
151958
How to show $f'(0)$ exist and is equal to $1$?
8836
If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
154612
$\int_{0}^{x} f(t) dt \ge {f(x)}$ holds or not on $[0,1]$
49511
Do we have $\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$?
1626
How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$
98344
Prove that $\int_0^1[f''(x)]^2dx\ge4.$
85750
If $f(x)=\int_{x-1}^x f(s)ds$, is $f$ constant? Periodic?
151519
If $f(f(x))=x^2-x+1$, what is $f(0)$?
89537
$ \int_0^x f(t)dt=\int_0^{ax}f(t)dt+ \int_0^{bx}f(t)dt$ implies $f$ constant
72660
How can I show that $f$ must be zero if $\int fg$ is always zero?
20021
Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$
145573
How to show that $f$ is a zero function?
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
101267
Show: $\int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)>t\right\})\, dt$
69301
If $\int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then every line $y=mx$ Intersect $ y^2+\int_{0}^{x}f(t)dt=2$
70790
Evaluation of $\lim_{q \to 0} (\int_0^1 |f|^q)^{1/q} $ for $f \in L^1([0,1])$.