query: Existence of $c \in [a,b]$ Such That $\int_{c-a}^{b-c}f(x)dx=0$
positive_passages.docid: 149399
positive_passages.text: Given continuous $f$ and $g$, there exists $c \in (a,b)$ such that $\int_{a}^{c} f(x) \ \mathrm dx = \int_{c}^{b} g(x) \ \mathrm dx$
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 68409
negative_passages.text: Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
negative_passages.docid: 6186
negative_passages.text: An Integrable function which is not essentially bounded on any subinterval of $[0,1]$
negative_passages.docid: 61799
negative_passages.text: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$
negative_passages.docid: 105769
negative_passages.text: Prove that $\exists a, b\in(0,1)$ such that $\int_0^{a} xf(x)dx=0\text{ and }\int_0^bxf(x)dx=\frac{b^2f(b)}{2}.$
negative_passages.docid: 73594
negative_passages.text: Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$
negative_passages.docid: 96050
negative_passages.text: $f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant
negative_passages.docid: 95469
negative_passages.text: 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 $x$
negative_passages.docid: 30694
negative_passages.text: Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
negative_passages.docid: 90369
negative_passages.text: Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function?
negative_passages.docid: 158180
negative_passages.text: An inequality of $\int_0^1 |f(x)|dx$
negative_passages.docid: 153886
negative_passages.text: Existence of antiderivative on a part $I$ of $\mathbb{R}$
negative_passages.docid: 26754
negative_passages.text: Find all continuous functions satisfying $\int_0^xf=(f(x))^2+C$ for some constant $C \neq 0$.
negative_passages.docid: 5180
negative_passages.text: How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
negative_passages.docid: 50012
negative_passages.text: $f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$