How many continuous functions exist such that $\int_{0}^{1} f(x)(1 − f(x))dx = \frac{1}4$?
112636
How many continuous function $f(x)$ exist such that $\int_{0}^{1}f(x)\big(1-f(x)\big)\mathrm dx = \frac{1}{4}$?
82280
Problem about continuous functions and the intermediate value theorem
145233
How to show $\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$ for continuous $f$?
53519
Finding $\int_0^1{\frac{x^4(1-x)^4}{1+x^2}}dx$
41201
How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.
124997
finding examples for a non negative and continuous function for which the infinite integral is finite but the limit at infinity doesn't exist
53726
Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$
134195
Holomorphic function $f:B(0,1) \to B(1,4)$ with $f(0)=3$ and $f(\frac{1}{2})=1$
21301
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
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 $x$
104510
For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
102657
If $f$ is continuous real function such that $f(11) = 10$ and $\forall x, f(x)f(f(x)) = 1$ then find $f(9)$
139744
For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$
98344
Prove that $\int_0^1[f''(x)]^2dx\ge4.$
115878
Continuous unbounded but integrable functions