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1: 1.4 Calculus of One Variable
§1.4(i) Monotonicity
If f ( x 1 ) 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 ( x ) 0 ( 0 ) ( = 0 ) for all x ( a , b ) , then f is nondecreasing (nonincreasing) (constant) on ( a , b ) . … In this case, g ( x ) = 𝒱 a , x ( f ) and h ( x ) = 𝒱 a , x ( f ) - 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 ) d x may be replaced in (18.2.1) by a positive measure d α ( x ) , where α ( x ) is a bounded nondecreasing function on the closure of ( a , b ) with an infinite number of points of increase, and such that 0 < a b x 2 n d α ( x ) < for all n . …
3: 2.8 Differential Equations with a Parameter
The regions of validity Δ j ( α j ) comprise those points ξ that can be joined to α j in Δ by a path 𝒬 j along which v is nondecreasing ( j = 1 ) or nonincreasing ( j = 2 ) as v passes from α j to ξ . …