Chapter1Functions and LimitsS1.10Properties of Continuous Functions
Chapter 1 Functions and Limits §1.10 Properties of Continuous Functions
I. Max-Min Theorem1.Definion:Let I, the domain of f (x), contain the point xo.f(x, )is a maximum value of f(x)on I if f(x.)≥ f(x)for all x in I; f(x)is a minimum value of f(x)on I if f(x)≤ f(x)for all x in I;2. TheoremIf f (x) is continuous on a closed interval [a,b], then f (x)attains both a maximum and a minimum value there.If f(x) is continuous on a closed interval [a,bl, thenis f(x) bounded on [a,b].S1.10 Properties of Continuous Functions
§1.10 Properties of Continuous Functions I. Max-Min Theorem 1.Definion: Let I, the domain of f (x), contain the point x0 . ◼ ( ) is a maximum value of on I if for all x in I; x0 f f (x) ( ) ( ) 0 f x f x ◼ ( ) is a minimum value of on I if for all x in I; x0 f f (x) ( ) ( ) f x0 f x 2. Theorem If f (x) is continuous on a closed interval [a,b], then f (x) attains both a maximum and a minimum value there. If is continuous on a closed interval [a,b], then is f (x) bounded on [a,b]. f (x)
I.Max-Min TheoremNOTES2.Continuity1.Closedinterval-x+1,0≤x<111,x=1f(x)=^(0,a)f(x) =x-x+3,1<x≤202a01S1.10 Properties of Continuous Functions
§1.10 Properties of Continuous Functions 1. Closed interval (0, ) 1 ( ) a x f x = 2. Continuity I. Max-Min Theorem NOTES 0 a − + = − + = 3,1 2 1, 1 1, 0 1 ( ) x x x x x f x 0 1 2
Il.Zero Theorem1.DefinitionIf f(x,)= 0, then x,is calleda zero of f(x) .a0ba21S1.10 Properties of Continuous Functions
§1.10 Properties of Continuous Functions II. Zero Theorem 1.Definition If , then is called a zero of . ( ) 0 f (x) f x0 = x0 0 a b 0 b a 0 a b 0 b a x y 0 b a x y
Il. Zero Theorem2. TheoremIf f(x) is continuous on a closed interval [a,b], andf (a)f (b) < 0, then f(x)has at least one zero on (a,b) (a,b), s.t. f() = 0Note:Sufficientbut not necessary-00bxbx1S1.10 Properties of Continuous Functions
§1.10 Properties of Continuous Functions II. Zero Theorem 2. Theorem If is continuous on a closed interval [a,b], and f (a) f (b) < 0, then f (x) has at least one zero on (a,b) . f (x) (a,b), s.t. f ( ) = 0 Note：Sufficient but not necessary 0 b a x y 0 b a x y