query: Why a continuous function cannot map rational numbers to irrational numbers
positive_passages.docid: 132478
positive_passages.text: Prove that continuous functions mapping irrationals to rationals must be constant
negative_passages.docid: 4381
negative_passages.text: Proof that the set of irrational numbers is dense in reals
negative_passages.docid: 117366
negative_passages.text: Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?
negative_passages.docid: 143174
negative_passages.text: property of continuous function
negative_passages.docid: 120935
negative_passages.text: Why is every continuous function on the reals determined by its value on rationals?
negative_passages.docid: 24817
negative_passages.text: Prove continuity on a function at every irrational point and discontinuity at every rational point.
negative_passages.docid: 24914
negative_passages.text: Does every continuous map from $\mathbb{Q}$ to $\mathbb{Q}$ extends continuously as a map from $\mathbb{R}$ to $\mathbb{R}$?
negative_passages.docid: 90745
negative_passages.text: Why aren’t continuous functions defined the other way around?
negative_passages.docid: 151427
negative_passages.text: f, g continuous for all rationals follow by continuous on all reals?
negative_passages.docid: 81428
negative_passages.text: Why does proof by contradiction not work on Rational numbers but only on Irrational numbers?
negative_passages.docid: 126842
negative_passages.text: Fixed point in a continuous function
negative_passages.docid: 41767
negative_passages.text: Sequence of rational numbers converging to an arbitrary irrational number $\beta$.
negative_passages.docid: 57233
negative_passages.text: Proof of continuity of Thomae Function at irrationals.
negative_passages.docid: 139763
negative_passages.text: A function takes every function value twice - proof it is not continuous
negative_passages.docid: 120947
negative_passages.text: continuous real-valued function
negative_passages.docid: 101319
negative_passages.text: Properties of a continuous map $f : \mathbb{R}^2\rightarrow \mathbb{R}$ with only finitely many zeroes
negative_passages.docid: 33330
negative_passages.text: Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$