# Questions tagged [bounded-variation]

For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0<x_1<\ldots<x_n=b$, $n\in\mathbb N^*$, then we have $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|\leq M$. This concept can be generalized to infinite intervals, requiring that the constant is uniform.

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### Estimating integral expression by using total variation of an integrand term

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/3} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y$$ for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
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### Why can we assume WLOG $\alpha$ is increasing?

I have a question regarding the proof of Theorem 9.8 from Mathematical Analysis by Tom Apostol below: Theorem 9.8: Let $\alpha$ be of a bounded variation on $[a,b]$. Assume that each term of the ...
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### Outward pointing vector on Lipschitz boundary

I have some questions on Lipschitz domains and their unit outward pointing vectors. My questions are listed below, I would appreciate direct answers and/or references on the subject. What is the good ...
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### A continuous bounded variation function in $[0,1]$ that is absolutely continuous in $(a,1]$, but is not in $[0,1]$
I am seeking for a continuous of bounded variation function in $[0,1]$ that is absolutely continuous in $(a,1]$ for all $a\in(0,1)$, but is not in $[0,1]$. The function $x\sin\left(\frac{1}{x}\right)$,...