show that $f$ and $g$ are constant functions in a region D
10194
If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
40609
Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
149399
Given continuous $f$ and $g$, there exists $c \in (a,b)$ such that $\int_{a}^{c} f(x) \ \mathrm dx = \int_{c}^{b} g(x) \ \mathrm dx$
17343
Function on $[a,b]$ that satisfies a Hölder condition of order $\alpha > 1 $ is constant
75479
Proving that this function is continuous on $G\times G$
40208
Locally Constant Functions on Connected Spaces are Constant
84313
$|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
157426
Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$
6450
Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
158407
derivative bounded by a constant multiple of the function
82001
How to prove the function is a constant
28315
Monotone functions and non-vanishing of derivative
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
9451
Show bounded and convex function on $\mathbb R$ is constant
28399
Showing $f$ is constant