query: If $f$ takes Cauchy sequence to Cauchy sequence then $f$ is continuous
positive_passages.docid: 12997
positive_passages.text: Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?
negative_passages.docid: 118332
negative_passages.text: proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
negative_passages.docid: 106060
negative_passages.text: If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
negative_passages.docid: 15579
negative_passages.text: Show that $f$ is uniformly continuous if limit exists
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: 55605
negative_passages.text: If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
negative_passages.docid: 150554
negative_passages.text: proof that $f'$ is continuous at $a$
negative_passages.docid: 129073
negative_passages.text: If $f'$ is bounded, then $f$ is uniformly continuous
negative_passages.docid: 94142
negative_passages.text: Proving that product of two Cauchy sequences is Cauchy
negative_passages.docid: 44402
negative_passages.text: If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
negative_passages.docid: 30965
negative_passages.text: $f$ is continuous on $X$ iff $f$ is continuous on every compact subset
negative_passages.docid: 93461
negative_passages.text: Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous.
negative_passages.docid: 14168
negative_passages.text: Prove the following sequence is a Cauchy Sequence
negative_passages.docid: 28730
negative_passages.text: product of two uniformly continuous functions is uniformly continuous
negative_passages.docid: 132013
negative_passages.text: How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
negative_passages.docid: 37966
negative_passages.text: If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
negative_passages.docid: 108696
negative_passages.text: If $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence, then $(a_n ^2)_{n\in\mathbb{N}}$ is also a Cauchy sequence