query: About the definition of continuous functions
positive_passages.docid: 157821
positive_passages.text: Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?
negative_passages.docid: 34344
negative_passages.text: Is a function that maps every compact set to a compact set continuous?
negative_passages.docid: 12107
negative_passages.text: How is the epsilon-delta definition of continuity equivalent to the following statement?
negative_passages.docid: 7876
negative_passages.text: If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
negative_passages.docid: 98772
negative_passages.text: Continuity of a function in $\mathbb{R}$
negative_passages.docid: 33289
negative_passages.text: Negation of definition of continuity
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: 115878
negative_passages.text: Continuous unbounded but integrable functions
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: 12523
negative_passages.text: Supremum of a Continuous Function is Continuous
negative_passages.docid: 9320
negative_passages.text: Extension of continuous function on a closed subset of $\mathbb{R}$
negative_passages.docid: 60700
negative_passages.text: Continuity of a function to the integers
negative_passages.docid: 15487
negative_passages.text: Continuous function from a connected set?
negative_passages.docid: 73585
negative_passages.text: Why is the function continuous, if it is defined only on integers?
negative_passages.docid: 50725
negative_passages.text: Inverse of function is continuous or not?
negative_passages.docid: 143618
negative_passages.text: If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)
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 $X\times Y$