Symmetric functional equation $ x f ( y ) + y f ( x ) = ( x + y ) f \left( x ^ 2 + y ^ 2 \right) $ on $ \mathbb N _ 0 $: how to show $ f$ is constant?
1071
Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)
82001
How to prove the function is a constant
153942
Solution of functional equation $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$
6680
Solve for all possible functions f: $|f(x)-f(y)|=2|x-y|$.
4178
Solve $f(x)f(2x^2) = f(2x^3+x)$
28399
Showing $f$ is constant
149354
Proof of continuity $f(x+y) = f(x) + f(y)$
49211
Solving $f(x+y) = f(x)f(y)f(xy)$
142873
What does the function f: x ↦ y mean?
151519
If $f(f(x))=x^2-x+1$, what is $f(0)$?
25953
Solving a functional relation $f\left( x \cdot f(y)\right)=x^2 \cdot y^a$
27809
Find the function equation $f(x+f(y)+yf(x))=y+f(x)+xf(y)$
71049
Functional equation $f(xy)=f(x)+f(y)$ and continuity
111556
Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?
148227
Rewriting of functional equation $f(xy)=f(x)+f(y)$
131586
Hard functional equation: $ f \big ( x y + f ( x ) \big) = f \big( f ( x ) f ( y ) \big) + x $
34343
Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $f(f(x) + y) = f(x^2 - y) + 4yf(x)$.