nondecreasing

(0.001 seconds)

3 matching pages

… ►

►If

f(x_{1})\leq f(x_{2})

for every pair

x_{1}

x_{2}

in an interval

I

such that

x_{1}<x_{2}

, then

f(x)

is nondecreasing on

I

. … ►If

f^{\prime}(x)\geq 0

(

\leq 0

) (

=0

) for all

x\in(a,b)

, then

f

is nondecreasing (nonincreasing) (constant) on

(a,b)

. … ►In this case,

g(x)=\mathcal{V}_{a,x}\left(f\right)

and

h(x)=\mathcal{V}_{a,x}\left(f\right)-f(x)

are nondecreasing bounded functions and

f(x)=g(x)-h(x)

. …

… ►More generally than (18.2.1)–(18.2.3),

w(x)\,\mathrm{d}x

may be replaced in (18.2.1) by a positive measure

\,\mathrm{d}\alpha(x)

, where

\alpha(x)

is a bounded nondecreasing function on the closure of

(a,b)

with an infinite number of points of increase, and such that

0<\int_{a}^{b}x^{2n}\,\mathrm{d}\alpha(x)<\infty

for all

n

. …

… ►The regions of validity

\mathbf{\Delta}_{j}(\alpha_{j})

comprise those points

\xi

that can be joined to

\alpha_{j}

\mathbf{\Delta}

by a path

\mathscr{Q}_{j}

along which

\Re v

is nondecreasing

(j=1)

or nonincreasing

(j=2)

v

passes from

\alpha_{j}

\xi

. …