# nondecreasing

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## 4 matching pages

##### 1: 1.4 Calculus of One Variable

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###### §1.4(i) Monotonicity

►If $f({x}_{1})\le f({x}_{2})$ for every pair ${x}_{1}$, ${x}_{2}$ in an interval $I$ such that $$, then $f(x)$ is*nondecreasing*on $I$. … ►If ${f}^{\prime}(x)\ge 0$ ($\le 0$) ($=0$) for all $x\in (a,b)$, then $f$ is nondecreasing (nonincreasing) (constant) on $(a,b)$. … ►A generalization of the Riemann integral is the*Stieltjes integral*${\int}_{a}^{b}f(x)d\alpha (x)$, where $\alpha (x)$ is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $d\alpha (x)$ is the*Stieltjes measure*. … ►For $\alpha (x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $d\alpha $ is*absolutely continuous*if $\alpha (x)$ is continuous and there exists a*weight function*$w(x)\ge 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that …##### 2: 1.16 Distributions

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►More generally, for $\alpha :[a,b]\to [-\mathrm{\infty},\mathrm{\infty}]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu}_{\alpha}$ (see §1.4(v)) can be considered as a distribution:
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##### 3: 2.8 Differential Equations with a Parameter

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►The regions of validity ${\mathbf{\Delta}}_{j}({\alpha}_{j})$ comprise those points $\xi $ that can be joined to ${\alpha}_{j}$ in $\mathbf{\Delta}$ by a path ${\mathcal{Q}}_{j}$ along which $\mathrm{\Re}v$ is nondecreasing
$(j=1)$ or nonincreasing $(j=2)$ as $v$ passes from ${\alpha}_{j}$ to $\xi $.
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##### 4: 18.2 General Orthogonal Polynomials

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►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by $d\mu (x)$, where the measure $\mu $ is the Lebesgue–Stieltjes measure ${\mu}_{\alpha}$ corresponding to a bounded nondecreasing function $\alpha $ on the closure of $(a,b)$ with an infinite number of points of increase, and such that $$ for all $n$.
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