query: If the graph of $f$ is closed, $f$ is continuous.
positive_passages.docid: 27661
positive_passages.text: The closed graph theorem in topology
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: 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: 122086
negative_passages.text: continuous function: preimage of closure of a set is closure of preimage
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: 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: 10747
negative_passages.text: Given function $g$ continuous in line $[0,\infty)$...
negative_passages.docid: 145919
negative_passages.text: If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
negative_passages.docid: 155923
negative_passages.text: continuity of $f(x,y)$ in $\mathbb R^2$.
negative_passages.docid: 57023
negative_passages.text: If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
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: 91682
negative_passages.text: Graph of a continuous function is closed
negative_passages.docid: 100740
negative_passages.text: proof of that a continuous function has a fixed point
negative_passages.docid: 86472
negative_passages.text: $f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ continuous ?
negative_passages.docid: 118207
negative_passages.text: Prove that the product of two continuous functions is continuous
negative_passages.docid: 53627
negative_passages.text: Proving continuity of $f$