Question

Is there any non-constant function $f(x)$ satisfying $f(x) f(y) = f(x) + f(y)$?

score 142 · Answer 1 · 2026-04-15 14:11:02

142

Views

Is there any non-constant function $f(x)$ satisfying $f(x) f(y) = f(x) + f(y)$?

Published on 15 Apr 2026 - 14:11

score 445 · Answer 2 · 2026-04-13 23:57:00

445

Views

Does $f(x+1)-f(x)$ constant imply $f$ is linear?

Published on 13 Apr 2026 - 23:57

score 180 · Answer 3 · 2026-04-16 22:52:12

Find $f: \mathbb{N_0} \to \mathbb{N_0}$ which satisfies $f^n(x+f(y))=f^{n+1}(x)+f^n(y) \text{ for } n \in \mathbb{N}.$

score 92 · Answer 4 · 2026-04-14 02:41:48

92

Views

$\frac{g(x,y)}{f(x)-f(y)}$ is increasing with $x$ and decreasing with $y$. Then, what is $g$?

Published on 14 Apr 2026 - 2:41

score 111 · Answer 5 · 2026-04-17 09:55:24

111

Views

Find all functions that satisfy $f(x^2f(y)^2)=f(x)^2f(y)$.

Published on 17 Apr 2026 - 9:55

score 213 · Answer 6 · 2026-04-17 01:04:32

213

Views

If $g(x+y)\le2\cdot\max\{g(x),g(y)\}$ and $g(xy)=g(x)g(y)$, show that $g(x+y)\le g(x)+g(y)$.

Published on 17 Apr 2026 - 1:04

score 116 · Answer 7 · 2026-04-17 04:40:48

116

Views

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(f(x)+xf(y))=x+yf(x).$

Published on 17 Apr 2026 - 4:40

score 57 · Answer 8 · 2026-04-14 23:20:06

57

Views

Find the smallest value at x=25

Published on 14 Apr 2026 - 23:20

score 443 · Answer 9 · 2026-04-16 16:24:00

443

Views

Find function $ f $ such that $f(\frac{x-3}{x+1})+f(\frac{x+3}{1-x})=x$

Published on 16 Apr 2026 - 16:24

score 92 · Answer 10 · 2026-04-15 03:55:09

92

Views

$f(a)+f(-b)=0$ implies $f(-a)+f(b)=0$. Then $f$ is an odd function?

Published on 15 Apr 2026 - 3:55

score 52 · Answer 11 · 2026-04-16 09:13:19

52

Views

For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)\geq 0$, we have: $2f(-a)+f(b)\leq 0$. Then, $f$ must be an odd function?

Published on 16 Apr 2026 - 9:13

score 63 · Answer 12 · 2026-04-08 17:39:47

63

Views

Solving Functional Equation from an Ordinary Generating Function

Published on 08 Apr 2026 - 17:39

score 84 · Answer 13 · 2026-04-14 00:37:15

84

Views

Is there any characterization for the real-valued function such that $y - x = y' - x' \Rightarrow f(y) - f(x) = f(y') - f(x')$

Published on 14 Apr 2026 - 0:37

score 47 · Answer 14 · 2026-04-12 00:55:55

47

Views

Why is injectiveness needed to show that a bounded linear operator is a closed map?

Published on 12 Apr 2026 - 0:55

score 232 · Answer 15 · 2026-04-16 21:36:07

232

Views

if $f(f(x+y))=f(x^2)+f(y^2)$ then $f(x)=?$

Published on 16 Apr 2026 - 21:36

List Question

Is there any non-constant function $f(x)$ satisfying $f(x) f(y) = f(x) + f(y)$?

Does $f(x+1)-f(x)$ constant imply $f$ is linear?

Find $f: \mathbb{N_0} \to \mathbb{N_0}$ which satisfies $f^n(x+f(y))=f^{n+1}(x)+f^n(y) \text{ for } n \in \mathbb{N}.$

$\frac{g(x,y)}{f(x)-f(y)}$ is increasing with $x$ and decreasing with $y$. Then, what is $g$?

Find all functions that satisfy $f(x^2f(y)^2)=f(x)^2f(y)$.

If $g(x+y)\le2\cdot\max\{g(x),g(y)\}$ and $g(xy)=g(x)g(y)$, show that $g(x+y)\le g(x)+g(y)$.

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(f(x)+xf(y))=x+yf(x).$

Find the smallest value at x=25

Find function $ f $ such that $f(\frac{x-3}{x+1})+f(\frac{x+3}{1-x})=x$

$f(a)+f(-b)=0$ implies $f(-a)+f(b)=0$. Then $f$ is an odd function?

For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)\geq 0$, we have: $2f(-a)+f(b)\leq 0$. Then, $f$ must be an odd function?

Solving Functional Equation from an Ordinary Generating Function

Is there any characterization for the real-valued function such that $y - x = y' - x' \Rightarrow f(y) - f(x) = f(y') - f(x')$

Why is injectiveness needed to show that a bounded linear operator is a closed map?

if $f(f(x+y))=f(x^2)+f(y^2)$ then $f(x)=?$

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