Find all functions $f:\mathbb N_0\to \mathbb N_0$ such that $f(a^2+b^2)=f(a)^2+f(b)^2$
32525
If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.
144090
How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$
23186
$f: \mathbb{R^2} \to \mathbb{R}$ a $C^\infty$ function such that $f(x,0)=f(0,y)=0$ then exists $g$ such that $f(x,y)=xy\, g(x,y)$
86128
Find all functions $ f : \Bbb R \to \Bbb R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y ∈ \Bbb R$.
73519
Determine all functions satisfying $f(x + f(x + y)) + f(xy) = x + f(x + y) +yf(x)$
29403
Finding functions $f: \Bbb R_*^+ \to \Bbb R_*^+$ with certain properties
47135
Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
5419
Continuous functions with $f^2(x)=g^2(x)\neq 0$
47856
$f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
68409
Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
62709
Functions that are sets of all function - proofs
117848
Find all entire functions
62966
Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$
53168
Surjective, Injective, Bijective Functions from $\mathbb Z$ to itself
29647
Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?
89257
Give an example of a function $f:\mathbb{R}^2 \mapsto \mathbb{R}^2$ such that