For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
95469
Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
95429
If $f$ is continuous with $ \int_0^{\infty}f(t)\,dt<\infty$ then which are correct?
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
67565
If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$
50103
Show that a continuous function satisfying $f(2^x)=f(3^x)$ for $x\in (0,\infty)$ is a constant function
97256
For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
72847
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
33989
If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
28289
Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
31319
An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.
109099
Prove that $\lim_{h \to 0}\frac{1}{h}\int_0^h{\cos{\frac{1}{t}}dt} = 0$
74583
Prove that $f'(0)=L$.
138054
Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
18261
Prove $\int_{a}^{b}{f(x)dx}>0$.