Question

If $f:[0,1] \to \mathbb{R}$ is of bounded variation, is $|f'|$ is integrable?

score 183 · Answer 1 · 2026-02-23 13:03:44

183

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If $f:[0,1] \to \mathbb{R}$ is of bounded variation, is $|f'|$ is integrable?

Published on 23 Feb 2026 - 13:03

score 123 · Answer 2 · 2023-02-18 05:55:02

123

Views

(1st) Stein's Real Analysis ch3 theorem 3.4, lemma 3.5, and Corollary 3.6

Published on 18 Feb 2023 - 5:55

score 145 · Answer 3 · 2023-02-18 09:29:15

145

Views

(2nd) Stein's Real Analysis ch3 theorem 3.4, lemma 3.5, and Corollary 3.6

Published on 18 Feb 2023 - 9:29

score 135 · Answer 4 · 2026-03-25 19:16:14

135

Views

Sufficient condition for a set to be a Reduced Boundary

Published on 25 Mar 2026 - 19:16

score 221 · Answer 5 · 2023-02-24 07:47:29

221

Views

If $f \in \mathcal{C}[a,b]$ and $|f| \in \mathrm{BV}[a,b]$, then how may we show $f \in \mathrm{BV}[a,b]$?

Published on 24 Feb 2023 - 7:47

score 199 · Answer 6 · 2023-03-07 09:41:36

199

Views

Showing that $f \in \mathrm{BV}[a,b] \implies \int_a^b |f'| \le V[f;a,b]$

Published on 07 Mar 2023 - 9:41

score 109 · Answer 7 · 2023-03-08 16:22:22

109

Views

The jumps of the total variation of a càdlàg function with finite variation.

Published on 08 Mar 2023 - 16:22

score 127 · Answer 8 · 2026-03-25 19:06:08

127

Views

Are two Riemann integrable functions, Riemann stieltjes integrable with respect ro each other?

Published on 25 Mar 2026 - 19:06

score 34 · Answer 9 · 2026-02-23 18:46:13

34

Views

Show that $g \in C([a,b]) \cap BV([a,b])$ with $\mu(E) = 0 \implies \mu(g(E)) = 0$ is in $AC([a,b])$

Published on 23 Feb 2026 - 18:46

score 85 · Answer 10 · 2023-03-30 07:33:17

85

Views

Sub-level sets of $BV$ functions - references and results

Published on 30 Mar 2023 - 7:33

score 131 · Answer 11 · 2023-04-08 16:33:15

Can we prove Weierstrass function is not of bounded variation according to the definition of bounded variation rather than its indifferentiability?

score 41 · Answer 12 · 2023-04-11 08:50:47

41

Views

Densitity of Locally Lipschitz functions on the space of locally integrable functions

Published on 11 Apr 2023 - 8:50

score 109 · Answer 13 · 2026-03-25 23:10:08

109

Views

Show that $\forall\varepsilon>0$, there exists a bounded continuous function $g(x)$ on $E$, such that $m(\{x\in E\mid f(x)\neq g(x)\})<\varepsilon$.

Published on 25 Mar 2026 - 23:10

score 45 · Answer 14 · 2023-04-15 09:11:40

45

Views

Variation of a Brownian Motion

Published on 15 Apr 2023 - 9:11

score 16 · Answer 15 · 2026-03-10 07:09:53

16

Views

Prove is an Outer Measure

Published on 10 Mar 2026 - 7:09

List Question

If $f:[0,1] \to \mathbb{R}$ is of bounded variation, is $|f'|$ is integrable?

(1st) Stein's Real Analysis ch3 theorem 3.4, lemma 3.5, and Corollary 3.6

(2nd) Stein's Real Analysis ch3 theorem 3.4, lemma 3.5, and Corollary 3.6

Sufficient condition for a set to be a Reduced Boundary

If $f \in \mathcal{C}[a,b]$ and $|f| \in \mathrm{BV}[a,b]$, then how may we show $f \in \mathrm{BV}[a,b]$?

Showing that $f \in \mathrm{BV}[a,b] \implies \int_a^b |f'| \le V[f;a,b]$

The jumps of the total variation of a càdlàg function with finite variation.

Are two Riemann integrable functions, Riemann stieltjes integrable with respect ro each other?

Show that $g \in C([a,b]) \cap BV([a,b])$ with $\mu(E) = 0 \implies \mu(g(E)) = 0$ is in $AC([a,b])$

Sub-level sets of $BV$ functions - references and results

Can we prove Weierstrass function is not of bounded variation according to the definition of bounded variation rather than its indifferentiability?

Densitity of Locally Lipschitz functions on the space of locally integrable functions

Show that $\forall\varepsilon>0$, there exists a bounded continuous function $g(x)$ on $E$, such that $m(\{x\in E\mid f(x)\neq g(x)\})<\varepsilon$.

Variation of a Brownian Motion

Prove is an Outer Measure

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