Question

Let $f$ be increasing and $x_1 < \dotsb < x_n$ in $[a,b]$. Show that $\sum_{i=1}^n o(f,x_i) < f(b)-f(a)$, where $o$ is the oscillation.

score 196 · Answer 1 · 2026-03-28 09:57:36

196

Views

Let $f$ be increasing and $x_1 < \dotsb < x_n$ in $[a,b]$. Show that $\sum_{i=1}^n o(f,x_i) < f(b)-f(a)$, where $o$ is the oscillation.

Published on 28 Mar 2026 - 9:57

score 1.5k · Answer 2 · 2017-04-11 01:20:41

1.5k

Views

Prove an inequality involving infimum and supremum

Published on 11 Apr 2017 - 1:20

score 162 · Answer 3 · 2026-03-28 13:13:59

162

Views

How to call a function on sets that preserves inclusion?

Published on 28 Mar 2026 - 13:13

score 85 · Answer 4 · 2026-03-28 14:56:50

85

Views

Let $Y = \{y_n\}$ be defined inductively by $y_1=1$ , $y_{n+1} = \frac 14\left(2y_n +3\right)$. Show that $\lim_{n\to \infty}y_n=\frac 32$

Published on 28 Mar 2026 - 14:56

score 2k · Answer 5 · 2026-03-28 14:55:08

2k

Views

Every Absolutely continuous function is of bounded variation and hence is differentiable almost everywhere.

Published on 28 Mar 2026 - 14:55

score 28 · Answer 6 · 2017-04-22 05:58:52

28

Views

Find a function satisfying given conditions

Published on 22 Apr 2017 - 5:58

score 91 · Answer 7 · 2017-04-24 18:28:08

91

Views

monotonicity of a univariate function

Published on 24 Apr 2017 - 18:28

score 358 · Answer 8 · 2017-04-26 09:30:06

358

Views

If $f:[0,1]\rightarrow [0,1]$ is increasing function. Suppose $0<f(0),f(1)<1$. Show $f$ has fixed point.

Published on 26 Apr 2017 - 9:30

score 105 · Answer 9 · 2017-04-27 20:46:06

105

Views

Let $f,g[0,1]\rightarrow \mathbb{R}$ be continuous. $\exists x_n$ $\forall n( g(x_n)=f(x_{n+1}))$. Prove $\exists w\in[0,1],f(w)=g(w)$

Published on 27 Apr 2017 - 20:46

score 548 · Answer 10 · 2017-04-30 12:46:28

548

Views

Continuous monotone maps and pre-image of connected sets.

Published on 30 Apr 2017 - 12:46

score 227 · Answer 11 · 2017-05-01 07:17:26

227

Views

Defining New Measure Using Monotone Function

Published on 01 May 2017 - 7:17

score 65 · Answer 12 · 2017-05-09 22:11:55

65

Views

$f(x)=x+2+2\ln x.$. Find the number of solutions of $f.$ Help?

Published on 09 May 2017 - 22:11

score 182 · Answer 13 · 2017-05-11 01:37:42

182

Views

Show monotonicity

Published on 11 May 2017 - 1:37

score 2.8k · Answer 14 · 2017-05-12 08:53:44

2.8k

Views

continuity of monotonically increasing function

Published on 12 May 2017 - 8:53

score 80 · Answer 15 · 2017-05-12 14:43:06

80

Views

$g_n$ is a sequence satifying$\int g_n d\mu <\frac{1}{n^2}$ for each $n\geq 1$. Prove that $\Sigma_{n=1}^{\infty}g_n(x) \leq + \infty$.

Published on 12 May 2017 - 14:43

List Question

Let $f$ be increasing and $x_1 < \dotsb < x_n$ in $[a,b]$. Show that $\sum_{i=1}^n o(f,x_i) < f(b)-f(a)$, where $o$ is the oscillation.

Prove an inequality involving infimum and supremum

How to call a function on sets that preserves inclusion?

Let $Y = \{y_n\}$ be defined inductively by $y_1=1$ , $y_{n+1} = \frac 14\left(2y_n +3\right)$. Show that $\lim_{n\to \infty}y_n=\frac 32$

Every Absolutely continuous function is of bounded variation and hence is differentiable almost everywhere.

Find a function satisfying given conditions

monotonicity of a univariate function

If $f:[0,1]\rightarrow [0,1]$ is increasing function. Suppose $0<f(0),f(1)<1$. Show $f$ has fixed point.

Let $f,g[0,1]\rightarrow \mathbb{R}$ be continuous. $\exists x_n$ $\forall n( g(x_n)=f(x_{n+1}))$. Prove $\exists w\in[0,1],f(w)=g(w)$

Continuous monotone maps and pre-image of connected sets.

Defining New Measure Using Monotone Function

$f(x)=x+2+2\ln x.$. Find the number of solutions of $f.$ Help?

Show monotonicity

continuity of monotonically increasing function

$g_n$ is a sequence satifying$\int g_n d\mu <\frac{1}{n^2}$ for each $n\geq 1$. Prove that $\Sigma_{n=1}^{\infty}g_n(x) \leq + \infty$.

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