query: Let $f: \Bbb R \to \Bbb R$ be such that $|f(x) -f(y)| \le |x - y |^2 \ \forall \ x,\ y \in \Bbb R$. How can I show that $f$ is constant?
positive_passages.docid: 23436
positive_passages.text: $f$ Holder continuous with Holder exponent $p>1\implies f \text{ is constant}$
negative_passages.docid: 50103
negative_passages.text: Show that a continuous function satisfying $f(2^x)=f(3^x)$ for $x\in (0,\infty)$ is a constant function
negative_passages.docid: 8854
negative_passages.text: Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$
negative_passages.docid: 54403
negative_passages.text: Given $f : \Bbb{R} \to \Bbb{R}$ is a continuous function. To prove that $\int_{0}^{1}f(x)x^2 dx = \frac{f(c)}{3}$ for some $c \in [0,1]$
negative_passages.docid: 36227
negative_passages.text: Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$
negative_passages.docid: 9451
negative_passages.text: Show bounded and convex function on $\mathbb R$ is constant
negative_passages.docid: 96539
negative_passages.text: is $f$ constant or dilation
negative_passages.docid: 84103
negative_passages.text: Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem
negative_passages.docid: 74950
negative_passages.text: Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
negative_passages.docid: 84510
negative_passages.text: If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
negative_passages.docid: 114245
negative_passages.text: What might I use to show that an entire function with positive real parts 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: 71857
negative_passages.text: Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
negative_passages.docid: 3818
negative_passages.text: Proving two entire functions are constant.
negative_passages.docid: 142494
negative_passages.text: Functions $f(x)/g(x), g(x)/h(x),h(x)/f(x)$ are constant
negative_passages.docid: 137043
negative_passages.text: To show that function is constant
negative_passages.docid: 101918
negative_passages.text: Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.