query: If $\int_a^b \left |{f^{\prime}}\right |=V\left. [ a,b \right ]$, $f$ of bounded variation, then $f $ absolutely continuous.
positive_passages.docid: 153957
positive_passages.text: Bounded variation and $\int_a^b |F'(x)|dx=T_F([a,b])$ implies absolutely continuous
negative_passages.docid: 50103
negative_passages.text: Show that a continuous function satisfying $f(2^x)=f(3^x)$ for $x\in (0,\infty)$ is a constant function
negative_passages.docid: 44574
negative_passages.text: Continuity of absolute value of a function
negative_passages.docid: 141366
negative_passages.text: Absolutely continuity and Lipschitz-like continuity
negative_passages.docid: 10898
negative_passages.text: Is an integral always continuous?
negative_passages.docid: 89568
negative_passages.text: Uniform continuity on $[a,b]$ and $ [b,c]$ $\implies$ uniform continuity on $[a,c]$.
negative_passages.docid: 117528
negative_passages.text: $f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous
negative_passages.docid: 91352
negative_passages.text: If $\Re(f)$ is bounded then f is constant.
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous
negative_passages.docid: 34836
negative_passages.text: Composition of Absolutely Continuous Functions
negative_passages.docid: 23436
negative_passages.text: $f$ Holder continuous with Holder exponent $p>1\implies f \text{ is constant}$
negative_passages.docid: 72240
negative_passages.text: Prove that if a linear operator is continuous, then it is bounded.
negative_passages.docid: 50366
negative_passages.text: Convolution is uniformly continuous and bounded
negative_passages.docid: 13732
negative_passages.text: $||f||_p\to ||f||_\infty$ under general assumptions
negative_passages.docid: 36166
negative_passages.text: Uniform continuity on (0,1) implies boundedness
negative_passages.docid: 102143
negative_passages.text: $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.
negative_passages.docid: 37850
negative_passages.text: Does the everywhere differentiability of $f$ imply it is absolutely continuous on a compact interval?