If $f(y)−f(x) \le (y−x)^2$ for all $x,y$, then $f$ is a constant function.
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
5416
How does one prove if a multivariate function is constant?
114826
Zero Partials imply Constant Function Theorem or Proof
142494
Functions $f(x)/g(x), g(x)/h(x),h(x)/f(x)$ are constant
56127
Prove a function is constant under certain conditions
44293
Functions satisfying $f\left( f(x)^2+f(y) \right)=xf(x)+y$
23842
$f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant?
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
113975
Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function
150502
If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$
153411
If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous
158647
If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
64650
If $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ then f is constant
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
82001
How to prove the function is a constant
113343
$f$ is constant if derivative equals zero