query: Let $f:S^1 \rightarrow \mathbb{R}$ be a continous function. Show that exists $x\in S^1$ such that $f(x)=f(-x)$
positive_passages.docid: 40209
positive_passages.text: $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
negative_passages.docid: 77323
negative_passages.text: Existence of continuous angle function $\theta:S^1\to\mathbb{R}$
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 95469
negative_passages.text: Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
negative_passages.docid: 134720
negative_passages.text: Show that there is no increasing function $f: \aleph_1 \to \mathbb{R}$
negative_passages.docid: 105652
negative_passages.text: Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
negative_passages.docid: 145233
negative_passages.text: How to show $\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$ for continuous $f$?
negative_passages.docid: 139744
negative_passages.text: For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$
negative_passages.docid: 108115
negative_passages.text: If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
negative_passages.docid: 143354
negative_passages.text: Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
negative_passages.docid: 1582
negative_passages.text: $f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges
negative_passages.docid: 13927
negative_passages.text: Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$
negative_passages.docid: 13575
negative_passages.text: Need help for proving that: $f(f^{-1}(A)) ⊆ A$
negative_passages.docid: 2717
negative_passages.text: Help proving existence of an specific $x_0$ for which $f''(x_0)=0$
negative_passages.docid: 143798
negative_passages.text: Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{[0,1]}$ almost everywhere.
negative_passages.docid: 8179
negative_passages.text: If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
negative_passages.docid: 71487
negative_passages.text: Real Analysis: Continuity of a Composition Function