If $\lvert f(x)-f(y) \rvert \le \lvert x-y \rvert ^2$ for all $x$ and $y$ in $\mathbb{R}$ then $f$ is a constant function
137043
To show that function is constant
5416
How does one prove if a multivariate function is constant?
105172
Prove that the entire function is constant over certain conditions.
63526
Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
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
98772
Continuity of a function in $\mathbb{R}$
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
27772
Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
132113
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
40208
Locally Constant Functions on Connected Spaces are Constant
58664
Proving a function is continuous for a fixed variable
106290
Prove a function is constant. (complex analysis)
58901
Why is every such function constant: $f(x)=f(x^2)$ for $x \in [0,1]$?
9451
Show bounded and convex function on $\mathbb R$ is constant
84206
How do I show that if $f$ is entire and $\{\lvert f(z)\rvert < M\}$ is connected for all $M$, then $f$ is a power function?
45222
When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?
26388
proof that function is constant