query: Proving that $f(x+y) = f(x)+f(y) $ and $f(xy) = f(x)f(y)$ imply $f(x) = 0$ or $f(x) = x$
positive_passages.docid: 13086
positive_passages.text: Overview of basic facts about Cauchy functional equation
negative_passages.docid: 129599
negative_passages.text: Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$
negative_passages.docid: 40609
negative_passages.text: Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
negative_passages.docid: 101394
negative_passages.text: Please explain the logic behind $d(xy) = y(dx) + x(dy)$
negative_passages.docid: 27809
negative_passages.text: Find the function equation $f(x+f(y)+yf(x))=y+f(x)+xf(y)$
negative_passages.docid: 31732
negative_passages.text: If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$
negative_passages.docid: 409
negative_passages.text: functions satisfying $f(x - y) = f(x) f(y) - f(a - x) f(a + y)$ and $f(0)=1$
negative_passages.docid: 32856
negative_passages.text: Functional equations $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$
negative_passages.docid: 77071
negative_passages.text: Solve the functional equation $f(xy) = f(x)f(y) - f(x + y) + 1$ and $f(1) = 2$
negative_passages.docid: 71857
negative_passages.text: Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
negative_passages.docid: 132093
negative_passages.text: Solve $f(f(x))=f(x)$
negative_passages.docid: 28088
negative_passages.text: When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$
negative_passages.docid: 86367
negative_passages.text: Computing $f(x)$ if $\int_1^{xy} f(t) dt = y \int_1^x f(t) dt + x\int_1^y f(t) dt$ and $f(1) = 3$
negative_passages.docid: 25467
negative_passages.text: Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
negative_passages.docid: 88683
negative_passages.text: Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$
negative_passages.docid: 2315
negative_passages.text: How to prove that $\lVert x + y\rVert = \lVert x\rVert + \lVert y \rVert \implies \lVert tx + (1-t)y\rVert = t\lVert x\rVert + (1-t)\lVert y \rVert$?
negative_passages.docid: 134043
negative_passages.text: Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $