Tagged Questions

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Examples of $f \in C^2[a,b]$ where the total variation of $f$ on $[a,x]$ is not in $C^2[a,b]$

Suppose $f:[a,b]\to\mathbb{R}$ is of bounded variation; define $V(x) = V[f;a,x]$ (the total variation of $f$ on $[a,x]$. I want to show that $V \in C^1[a,b]$. Since $f'$ is continuous, hence bounded ...
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Discontinuous local martingales with finite variations

I would like to know if a discontinuous local martingale with paths of finite variations almost surely is a martingale. I feel that it should be the case but can't find a straightforward argument. As ...
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Uniform limit of continuous functions bounded variation

Prove or disprove that if $f:[a,b]\rightarrow\mathbb{R}$ is the uniform limit of a sequence of continuous functions each of which is of bounded variation, then $f$ is of bounded variation on $[a,b].$
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Can I make a BV function right-continuous this way?

Math people: This question is related to how can you "fix" one of the definitions of a BV function of one variable? . Suppose $f \in BV([0,1])$. I really have two-three questions. The ...
Prove that $BV[a,b]$ is the smallest linear space containing all monotone functions on $[a,b].$
Let $f\colon[0,1] \rightarrow \mathbb R$ (all real numbers) be defined by $\displaystyle f(x) = x \sin \left(\frac{\pi}{2x}\right)$ if $0 \lt x \le 1$ and $f(x) = 0$ if $x=0$. Determine whether ...