Show that $f$ is locally constant function.
50115
Show that the holomorphic function $u + iv$ must be constant if $u = h \circ v$. Problem from Remmert.
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
58664
Proving a function is continuous for a fixed variable
113687
$f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
111458
Prove that $f$ is differentiable
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
105700
Proving $\,f$ is constant.
35787
An entire function with a bound on its integral is constant
103451
Show that $Y$ is locally compact.
133386
If $f$ is entire and $\lim_\limits{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant
32789
Show bounded harmonic function on $\mathbb{C}$ is constant.
80939
How to show that every $\alpha$-Hölder function, with $\alpha>1$, is constant?
82367
Prove that a holomorphic function with postive real part is constant
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
4171
If $f$ is locally Lipschitz on $X$ and $X$ is compact, then $f$ is Lipschitz on $X$.
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
113343
$f$ is constant if derivative equals zero