Rudin exercise: If $f$ is a diferentiable mapping on an open connected set $E$ and $f'(x) = 0$ for all $x \in E$, then $f$ is constant.
40208
Locally Constant Functions on Connected Spaces are Constant
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
118873
Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
153855
Complex analysis, showing a function is constant
158647
If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
77267
Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
8854
Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$
97316
Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
27772
Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
105068
Proving that the function is differentiable and $f'(0)=0$
49155
$f: \Omega \rightarrow \Omega$ holomorphic, $f(0) = 0$, $f'(0) = 1$ implies $f(z) = z$
34719
Show this holomorphic function is constant
96539
is $f$ constant or dilation