query: Existence of real valued function continuous at $\mathbb Q$ discontinuous at $\mathbb R\backslash \mathbb Q$
positive_passages.docid: 18222
positive_passages.text: Set of continuity points of a real function
negative_passages.docid: 141423
negative_passages.text: The continuity of $f(x,y) = \frac {\sin (xy)} {\sqrt {x^2 + y^2}}$ at $(0,0)$
negative_passages.docid: 129324
negative_passages.text: Let $f:[0;1]\to\mathbb R$ be continiously differentiable
negative_passages.docid: 105652
negative_passages.text: Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
negative_passages.docid: 58816
negative_passages.text: Existence of a continuous function which does not achieve a maximum.
negative_passages.docid: 83446
negative_passages.text: bijective continuous function on $\mathbb R^n$ not homeomorphism?
negative_passages.docid: 33330
negative_passages.text: Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
negative_passages.docid: 122160
negative_passages.text: $f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$
negative_passages.docid: 50427
negative_passages.text: Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?
negative_passages.docid: 118192
negative_passages.text: Is there a bijective function $f: \mathbb{R}\to\mathbb{R}$ that is discontinuous?
negative_passages.docid: 27800
negative_passages.text: For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
negative_passages.docid: 62817
negative_passages.text: Functions which are Continuous, but not Bicontinuous
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: 142459
negative_passages.text: Continuity of a Function in Terms of Closure and Interior
negative_passages.docid: 74950
negative_passages.text: Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$