query: Prove that a linear map is continuous if and only if it is continuous at 0.
positive_passages.docid: 154521
positive_passages.text: Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
negative_passages.docid: 54222
negative_passages.text: Prove the map has a fixed point
negative_passages.docid: 154559
negative_passages.text: For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
negative_passages.docid: 72952
negative_passages.text: Uniqueness of a linear map on a basis of a vector space
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: 102457
negative_passages.text: for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
negative_passages.docid: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
negative_passages.docid: 97761
negative_passages.text: Show that a proper continuous map from $X$ to locally compact $Y$ is closed
negative_passages.docid: 64260
negative_passages.text: Prove that a function between metric spaces is continuous iff the inverse image of a closed set is closed
negative_passages.docid: 130299
negative_passages.text: Prove that if a function $f: X\to Y$ continuous then its graph is closed
negative_passages.docid: 28637
negative_passages.text: Proving continuous image of compact sets are compact
negative_passages.docid: 62887
negative_passages.text: Proof that a linear transformation is continuous
negative_passages.docid: 14578
negative_passages.text: Proving that an additive function $f$ is continuous if it is continuous at a single point
negative_passages.docid: 38033
negative_passages.text: A surjective linear map into a finite dimensional space is open
negative_passages.docid: 2198
negative_passages.text: Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous
negative_passages.docid: 8836
negative_passages.text: If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
negative_passages.docid: 83831
negative_passages.text: Continuity based on restricted continuity of two subsets