Let $f$ be a continuous real valued function on $[0,1]$ and $\int_0^1f(x)x^ndx = 0$ for $n = 0,1,2...$ Prove that $f$ is identically zero on $[0,1]$
12402
Showing a function on $[0,1]$ is the zero function
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
117649
$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent
76285
Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f $ is identically $0$.
104510
For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
76790
Condition to guarantee $f=0$ on $[a,b]$
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. $\int_0^1 f(x)dx=\int_0^1 xf(x)dx=1$
120774
Show that $f$ has exactly one zero on the square $Q =$ {$x + iy ∈ \Bbb C : |x| < 1, |y| < 1$}.
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
61782
How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
3518
Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous
134787
How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
72847
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
21954
A question about $f(x)\equiv 0$