$f(yf(x) + y) = xy + f(y)$ for all $x, y.$ Prove $f$ is surjective
31732
If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$
16092
How to prove that the composition of two surjective functions is surjective
72251
$g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective
155806
(Olympiad Question) $f(2x) + 2f(y) = f(f(x+y))$
88683
Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$
37976
Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$
121395
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
74502
Proving the bijection of a continuous function $f$ such that $|f(x)-f(y)| \geq \log(1+|x-y|) \forall x,y$ in $\mathbb{R}.$
48925
BMO1 2009/10 Q5 functional equation: $f(x)f(y) = f(x + y) + xy$
81720
$f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$
105455
Prove that "injective function $f:X\to Y$ exists" and "surjective function $g:Y\to X$ exists" is logically equivalent.
21891
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
81671
Proof of identity $\sqrt {xy} = \sqrt x \sqrt y$ for $x,y \in \mathbb R^+$
116056
Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
34693
Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
148227
Rewriting of functional equation $f(xy)=f(x)+f(y)$
109942
Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$