query: Proof that $ f: X \times \mathbb{R} \to X: (x,\lambda) \mapsto \lambda x$ is continuous
positive_passages.docid: 53850
positive_passages.text: Proof that every normed vector space is a topological vector space
negative_passages.docid: 140052
negative_passages.text: Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
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: 55238
negative_passages.text: Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
negative_passages.docid: 108057
negative_passages.text: If $(f_n)\to f$ uniformly and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous
negative_passages.docid: 39235
negative_passages.text: Argue that the iterated integral of a continuous function is continuous
negative_passages.docid: 145368
negative_passages.text: How to show $f:Y\to X^{+}$ is continuous?
negative_passages.docid: 97104
negative_passages.text: Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
negative_passages.docid: 125682
negative_passages.text: Proof verification: the product of two continuous functions is continuous
negative_passages.docid: 32359
negative_passages.text: Inner product of two continuous maps is continuous
negative_passages.docid: 31548
negative_passages.text: Proof of "every convex function is continuous"
negative_passages.docid: 85623
negative_passages.text: Proof of continuity implied by differentiability
negative_passages.docid: 52388
negative_passages.text: The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition
negative_passages.docid: 9320
negative_passages.text: Extension of continuous function on a closed subset of $\mathbb{R}$
negative_passages.docid: 19916
negative_passages.text: $f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
negative_passages.docid: 47
negative_passages.text: Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.
negative_passages.docid: 77621
negative_passages.text: Continuity of function mapping connected set to connected set