query: Is there any noncontinuous function f(x), such that the absolute value of f(x) is continuous?
positive_passages.docid: 145017
positive_passages.text: Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
negative_passages.docid: 13765
negative_passages.text: Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
negative_passages.docid: 36983
negative_passages.text: Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x?
negative_passages.docid: 15690
negative_passages.text: Is a function defined at a single point continuous?
negative_passages.docid: 111990
negative_passages.text: Continuous function from $(0, 1]$ onto $(0, 1)$?
negative_passages.docid: 52632
negative_passages.text: $f\colon(0,\infty)\to \mathbb R$ be continuous ; $f(x)\le f(nx) , \forall n \in \mathbb N , \forall x >0$ , then $\lim_{x\to \infty} f(x)$ exists?
negative_passages.docid: 33488
negative_passages.text: A uniformly continuous function such that the derivative is not bounded and is not defined on a compact?
negative_passages.docid: 25010
negative_passages.text: Argue that there exists a continuous function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies $g(x) = \frac{1-e^{-2x^2}}{x^2}$ for $x \neq 0$
negative_passages.docid: 24306
negative_passages.text: Is there a continuous function $f(x)$ such that the inverse function is $1/f(x)$?
negative_passages.docid: 2321
negative_passages.text: Can there be two distinct, continuous functions that are equal at all rationals?
negative_passages.docid: 141945
negative_passages.text: Why is $f(x) = \sqrt{x}$ is continuous at $x=0$? Does $\lim_{x \to 0^{-}} \sqrt{x}$ exist?
negative_passages.docid: 83338
negative_passages.text: Show that there's no continuous function that takes each of its values $f(x)$ exactly twice.
negative_passages.docid: 72847
negative_passages.text: Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
negative_passages.docid: 126759
negative_passages.text: Unbounded continuous function on non-compact metric space
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: 64487
negative_passages.text: Continuous function and dense set
negative_passages.docid: 158309
negative_passages.text: Find a function that is continuous only on rationals