query: $|f(x)-f(y)|\leqslant C|x-y|^{\alpha},\forall x,y\in(a,b)$ Prove that $f$ is constant in $(a,b)$.
positive_passages.docid: 17343
positive_passages.text: Function on $[a,b]$ that satisfies a Hölder condition of order $\alpha > 1 $ is constant
negative_passages.docid: 114488
negative_passages.text: If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
negative_passages.docid: 27772
negative_passages.text: Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
negative_passages.docid: 48592
negative_passages.text: Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
negative_passages.docid: 84510
negative_passages.text: If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
negative_passages.docid: 129716
negative_passages.text: Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
negative_passages.docid: 61799
negative_passages.text: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
negative_passages.docid: 73547
negative_passages.text: Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
negative_passages.docid: 63255
negative_passages.text: Proving an entire, complex function with below bounded modulus is constant.
negative_passages.docid: 3820
negative_passages.text: Prove that $||x|-|y||\le |x-y|$
negative_passages.docid: 22303
negative_passages.text: How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 28399
negative_passages.text: Showing $f$ is constant
negative_passages.docid: 82001
negative_passages.text: How to prove the function is a constant
negative_passages.docid: 20273
negative_passages.text: Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
negative_passages.docid: 105172
negative_passages.text: Prove that the entire function is constant over certain conditions.
negative_passages.docid: 111186
negative_passages.text: Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$