query: Showing that a function is continuous when restriction on closed set is
positive_passages.docid: 154559
positive_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: 145368
negative_passages.text: How to show $f:Y\to X^{+}$ is continuous?
negative_passages.docid: 9320
negative_passages.text: Extension of continuous function on a closed subset of $\mathbb{R}$
negative_passages.docid: 62817
negative_passages.text: Functions which are Continuous, but not Bicontinuous
negative_passages.docid: 71487
negative_passages.text: Real Analysis: Continuity of a Composition Function
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: 46287
negative_passages.text: Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$
negative_passages.docid: 93021
negative_passages.text: $f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
negative_passages.docid: 122086
negative_passages.text: continuous function: preimage of closure of a set is closure of preimage
negative_passages.docid: 100187
negative_passages.text: Show function is continuous at origin
negative_passages.docid: 53627
negative_passages.text: Proving continuity of $f$
negative_passages.docid: 14896
negative_passages.text: Prove: the set of zeros of a continuous function is closed.
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: 136851
negative_passages.text: Proof that the set of subsequential limits is closed
negative_passages.docid: 153730
negative_passages.text: Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
negative_passages.docid: 122190
negative_passages.text: Kernel of $T$ is closed iff $T$ is continuous
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