# nondecreasing

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## 3 matching pages

##### 1: 1.4 Calculus of One Variable
###### §1.4(i) Monotonicity
If $f(x_{1})\leq f(x_{2})$ for every pair $x_{1}$, $x_{2}$ in an interval $I$ such that $x_{1}, 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 $f(x)=g(x)-h(x)$. …
##### 2: 18.2 General Orthogonal Polynomials
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 $n$. …
##### 3: 2.8 Differential Equations with a Parameter
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 $\mathscr{Q}_{j}$ along which $\Re v$ is nondecreasing $(j=1)$ or nonincreasing $(j=2)$ as $v$ passes from $\alpha_{j}$ to $\xi$. …