How to prove this function is actually constant?
132113
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
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.
7878
Proving that a doubly-periodic entire function $f$ is constant.
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
145368
How to show $f:Y\to X^{+}$ is continuous?
75479
Proving that this function is continuous on $G\times G$
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
112211
Show an increasing function f from omega_1 to R is eventually constant
58901
Why is every such function constant: $f(x)=f(x^2)$ for $x \in [0,1]$?
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
48867
why all constant functions are continuous.$?$
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
77980
How to check the real analyticity of a function?
38885
Prove the function is continuous, exercise from Conway's "Functions of One Complex Variable I"
64650
If $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ then f is constant
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
27772
Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.