Suppose $f$ is a Lebesgue integrable function on$[0,1]$ which satisfies $\int x^k f(x) dx=0$ , prove $f=0 \text{ a.e.}$
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
154612
$\int_{0}^{x} f(t) dt \ge {f(x)}$ holds or not on $[0,1]$
5229
Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$.
2717
Help proving existence of an specific $x_0$ for which $f''(x_0)=0$
64571
Prove that $\int_{E}f =\lim \int_{E}f_{n}$
16109
Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
144589
Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
98344
Prove that $\int_0^1[f''(x)]^2dx\ge4.$
42853
If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$
104510
For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
28949
Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
130322
Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `
89537
$ \int_0^x f(t)dt=\int_0^{ax}f(t)dt+ \int_0^{bx}f(t)dt$ implies $f$ constant
117246
Show that $\lim\limits_{a \to 0^+} \int^1_0 ax^{a−1}f(x)dx=f(0)$.
63950
Prove that to each $\epsilon >0$, there exists a $\delta >0$ so that the Lebesgue integral...
118873
Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
124061
Prove the reciprocal of a function is integrable if it is bounded