# nondecreasing

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3 matching pages ♦

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## 3 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 $(a,b)$. … ►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 $f(x)=g(x)-h(x)$. …##### 2: 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 a positive measure $d\alpha (x)$, 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 $$ for all $n$.
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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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