query: Show that f is discontinuous at every rational and continuous at every irrational.
positive_passages.docid: 24817
positive_passages.text: Prove continuity on a function at every irrational point and discontinuity at every rational point.
negative_passages.docid: 18222
negative_passages.text: Set of continuity points of a real function
negative_passages.docid: 155923
negative_passages.text: continuity of $f(x,y)$ in $\mathbb R^2$.
negative_passages.docid: 126212
negative_passages.text: Proving uniform continuity on an interval
negative_passages.docid: 150235
negative_passages.text: $\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous
negative_passages.docid: 125682
negative_passages.text: Proof verification: the product of two continuous functions is continuous
negative_passages.docid: 158888
negative_passages.text: Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
negative_passages.docid: 19916
negative_passages.text: $f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
negative_passages.docid: 53627
negative_passages.text: Proving continuity of $f$
negative_passages.docid: 34940
negative_passages.text: Continuity of $f(x) = x^n$
negative_passages.docid: 16291
negative_passages.text: Discontinuous derivative.
negative_passages.docid: 77267
negative_passages.text: Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
negative_passages.docid: 61500
negative_passages.text: If f is continous and exist real numbers show that $f(x)=0$
negative_passages.docid: 92734
negative_passages.text: Existence of a sequence of continuous functions convergent pointwise to the indicator function of irrational numbers
negative_passages.docid: 19146
negative_passages.text: Showing that $f$ has exactly one fixed point
negative_passages.docid: 7358
negative_passages.text: For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.
negative_passages.docid: 151427
negative_passages.text: f, g continuous for all rationals follow by continuous on all reals?