Proof that $\int_{0}^{1} \frac{f(x)}{f(x) + f(1-x)}dx = 1/2$, where $f:[0,1]\rightarrow \mathbb{R}$ is strictly positive and continuous?
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
129551
Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
145233
How to show $\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$ for continuous $f$?
132203
What is the intuition behind why the integration of $f(x) = x$ for closed interval of negative to positive infinity diverges, rather than being zero?
109896
Proving that the function $f(x,y)=\frac{x^2y}{x^2 + y^2}$ with $f(0,0)=0$ is continuous at $(0,0)$.
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
94869
Show that $ \int_{k}^{2} \frac{1}{f(y)} =1$
66504
Proof that $f$ is Lebesgue-integrable and calculate its integral
142637
If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$
48314
Show that if a function is not negative and its integral is $0$ than the function is $0$
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
66758
$f_n:[0,1]\to [0,1]$ continuous, $f_n\to f$ uniformly, prove: ${\frac1n}\sum_{k=1}^{n}{f_k} \to f$ uniformly
54940
If integral is zero and function is continuous and non negative, then what about the function?
18261
Prove $\int_{a}^{b}{f(x)dx}>0$.
776
Entire function with positive real part is constant (no Picard)
17996
Integral $\int_0^1 \frac{\ln(x+\sqrt{1-x^2})}{x}dx$
16212
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$