query: functional equation $f\left(x\right)+a=f\left(x+b\right)$
positive_passages.docid: 144601
positive_passages.text: Given $f(x+T) = f(x) + a$, prove that $f(x) = \varphi(x) + {a \over T}x$, where $\varphi(x)$ is periodic with period $T$
negative_passages.docid: 117353
negative_passages.text: Polynomial equation $f(x)f(2x^2)=f(2x^3+x)$
negative_passages.docid: 69673
negative_passages.text: Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
negative_passages.docid: 76679
negative_passages.text: Prove there are no other solutions of functional equation $f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$
negative_passages.docid: 55693
negative_passages.text: The functional equation $ f \big( x - f ( y ) \big) = f \big( f ( y ) \big) + x f ( y ) + f ( x ) - 1 $
negative_passages.docid: 104209
negative_passages.text: Functional equation - Cyclic Substitutions
negative_passages.docid: 81799
negative_passages.text: Examples of functions where $f(ab)=f(a)+f(b)$
negative_passages.docid: 16109
negative_passages.text: Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
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 $f(c)=f(c+\frac{b-a}{2})$
negative_passages.docid: 61162
negative_passages.text: Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $
negative_passages.docid: 62966
negative_passages.text: Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$
negative_passages.docid: 37976
negative_passages.text: Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$
negative_passages.docid: 120707
negative_passages.text: Proof regard to the function $ f\left(x\right)=\begin{cases} x^{2}\sin\left(\frac{1}{x^{2}}\right) & x\neq0\\ 0 & x=0 \end{cases} $
negative_passages.docid: 69734
negative_passages.text: how to find all functions such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$
negative_passages.docid: 62111
negative_passages.text: Functional Equation $f(mn)=f(m)f(n)$.
negative_passages.docid: 28286
negative_passages.text: About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$
negative_passages.docid: 132093
negative_passages.text: Solve $f(f(x))=f(x)$