query: Is it true that there exists $c\in[a,b]$ such that $f(c)=\frac{Af(a)+Bf(b)}{A+B}$?
positive_passages.docid: 143174
positive_passages.text: property of continuous function
negative_passages.docid: 81799
negative_passages.text: Examples of functions where $f(ab)=f(a)+f(b)$
negative_passages.docid: 75266
negative_passages.text: $f(a+b)=f(a)+f(b)$ but $f$ is not linear
negative_passages.docid: 61186
negative_passages.text: For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$
negative_passages.docid: 47856
negative_passages.text: $f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
negative_passages.docid: 90565
negative_passages.text: 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 $f(x) = f(x+\frac{1}{n})$.
negative_passages.docid: 52672
negative_passages.text: Solution of $\frac{a}{b}=\frac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$
negative_passages.docid: 62770
negative_passages.text: Does $a\mid bc$ imply $a\mid b$ or $a\mid c$?
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: 18126
negative_passages.text: If $f \colon [a,b] \rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$
negative_passages.docid: 56413
negative_passages.text: $f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
negative_passages.docid: 35889
negative_passages.text: Suppose that f(0) = f(1) = 0 and x ∈ (0, 1). Prove that there exists c ∈ (0, 1) such that f'(c) = f(x).
negative_passages.docid: 92376
negative_passages.text: Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
negative_passages.docid: 155063
negative_passages.text: Is Cl(f(A)) a subset of f(Cl(A))?
negative_passages.docid: 30685
negative_passages.text: Is $f(a)\!=\!0\!=\!f(b)\Rightarrow (x\!-\!a)(x\!-\!b)\mid f(x)\,$ true if $\,a=b?\ $ [Double Factor Theorem]
negative_passages.docid: 24810
negative_passages.text: Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
negative_passages.docid: 77777
negative_passages.text: If $0\le a,b,c\le 1$, then $\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}\le 2$