query: Uniform continuity on bounded subset of $\mathbb{R}$ implies boundedness of the function
positive_passages.docid: 36149
positive_passages.text: A uniformly continuous function maps bounded set to bounded sets
negative_passages.docid: 64619
negative_passages.text: Continuity of a function of product spaces
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: 9320
negative_passages.text: Extension of continuous function on a closed subset of $\mathbb{R}$
negative_passages.docid: 148994
negative_passages.text: $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
negative_passages.docid: 60070
negative_passages.text: Unique uniformly continuous function into complete space
negative_passages.docid: 123981
negative_passages.text: Is it possible to write any bounded continuous function as a uniform limit of smooth functions
negative_passages.docid: 89568
negative_passages.text: Uniform continuity on $[a,b]$ and $ [b,c]$ $\implies$ uniform continuity on $[a,c]$.
negative_passages.docid: 47188
negative_passages.text: Uniformly Continuous Function sending Bounded Set to Unbounded One
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: 155618
negative_passages.text: Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$?
negative_passages.docid: 51156
negative_passages.text: Function is Continuous When Restricted to Closed Sets Which Cover the Space
negative_passages.docid: 31360
negative_passages.text: Hellinger-Toeplitz Theorem and Uniform Boundedness Principle
negative_passages.docid: 129757
negative_passages.text: Uniformly convergent implies equicontinuous
negative_passages.docid: 15284
negative_passages.text: $f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.
negative_passages.docid: 158344
negative_passages.text: Uniform continuous maps that converge uniformly to a function f. What can we say about f?
negative_passages.docid: 73594
negative_passages.text: Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$