Question

Let $f$ be a real-valued, everywhere differentiable function on $[0,1]$. Suppose $f \in BV[0,1]$. Prove $f \in AC[0,1]$.

score 91 · Answer 1 · 2026-02-23 17:26:27

91

Views

Let $f$ be a real-valued, everywhere differentiable function on $[0,1]$. Suppose $f \in BV[0,1]$. Prove $f \in AC[0,1]$.

Published on 23 Feb 2026 - 17:26

score 67 · Answer 2 · 2023-07-20 14:13:31

How to show $f(x) = \begin{cases} \frac{\sin(x)}{x}, & \text{when } x \neq 0,\\ 0, & \text{when } x = 0. \end{cases}$ is of bounded variation?

score 115 · Answer 3 · 2026-03-25 09:50:49

115

Views

What is the positive variation of a function?

Published on 25 Mar 2026 - 9:50

score 57 · Answer 4 · 2026-03-26 07:45:13

57

Views

Why if p-th variation is finite, then q-th variation is also finite for q>p?

Published on 26 Mar 2026 - 7:45

score 194 · Answer 5 · 2023-08-25 17:46:02

194

Views

$\sum_{n=1}^\infty 2^{-n} f_n$ has unbounded variation on every $[a,b] \subset [0,1]$ with $a < b$

Published on 25 Aug 2023 - 17:46

score 130 · Answer 6 · 2023-09-06 17:34:15

130

Views

Proof request: Extending vector valued function of unit outer normals to the whole space $\Bbb R^n$.

Published on 06 Sep 2023 - 17:34

score 175 · Answer 7 · 2023-09-08 13:50:10

175

Views

Approximating a function of bounded variation function by a stepfunction obtained by averaging over measurable sets

Published on 08 Sep 2023 - 13:50

score 102 · Answer 8 · 2023-09-16 09:55:17

102

Views

How pathological can a function of bounded variation be?

Published on 16 Sep 2023 - 9:55

score 119 · Answer 9 · 2023-10-20 09:48:40

119

Views

Bounded variation of functions and sequences

Published on 20 Oct 2023 - 9:48

score 83 · Answer 10 · 2026-03-26 03:06:09

83

Views

Naturalness of definition of line integral

Published on 26 Mar 2026 - 3:06

score 41 · Answer 11 · 2023-11-02 02:21:53

41

Views

Bounded Variation In Compact set

Published on 02 Nov 2023 - 2:21

score 67 · Answer 12 · 2023-11-02 03:11:51

67

Views

If $f$ is bounded variation then $|f|$ is bounded variation

Published on 02 Nov 2023 - 3:11

score 277 · Answer 13 · 2023-11-02 14:06:47

277

Views

Bounding $L^1$ norm of the difference between a function $f:\mathbb R^n\to\mathbb R$ of bounded variation and a piecewise constant approximation

Published on 02 Nov 2023 - 14:06

score 35 · Answer 14 · 2026-02-23 06:42:25

35

Views

For an arbitrary continuous function $f$, is the Stieltjes integral $\int_0^1(df(x))^3=0$?

Published on 23 Feb 2026 - 6:42

score 63 · Answer 15 · 2023-12-04 16:31:03

63

Views

Equality of continuous functions of bounded and unbounded variations

Published on 04 Dec 2023 - 16:31

List Question

Let $f$ be a real-valued, everywhere differentiable function on $[0,1]$. Suppose $f \in BV[0,1]$. Prove $f \in AC[0,1]$.

How to show $f(x) = \begin{cases} \frac{\sin(x)}{x}, & \text{when } x \neq 0,\\ 0, & \text{when } x = 0. \end{cases}$ is of bounded variation?

What is the positive variation of a function?

Why if p-th variation is finite, then q-th variation is also finite for q>p?

$\sum_{n=1}^\infty 2^{-n} f_n$ has unbounded variation on every $[a,b] \subset [0,1]$ with $a < b$

Proof request: Extending vector valued function of unit outer normals to the whole space $\Bbb R^n$.

Approximating a function of bounded variation function by a stepfunction obtained by averaging over measurable sets

How pathological can a function of bounded variation be?

Bounded variation of functions and sequences

Naturalness of definition of line integral

Bounded Variation In Compact set

If $f$ is bounded variation then $|f|$ is bounded variation

Bounding $L^1$ norm of the difference between a function $f:\mathbb R^n\to\mathbb R$ of bounded variation and a piecewise constant approximation

For an arbitrary continuous function $f$, is the Stieltjes integral $\int_0^1(df(x))^3=0$?

Equality of continuous functions of bounded and unbounded variations

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