《高等数学》课程电子教案（PPT课件）Chapter 1 Functions and Limits §1.9 Continuity of Functions

Chapter1Functions and LimitsS 1.9 Continuity of Functions

Chapter 1 Functions and Limits §1.9 Continuity of Functions

IntroductionI. Continuity at a PointI. Continuity on an IntervalIⅢI.Classification of Discontinuous PointsIV.Operations onContinuous FunctionsV.Continuity of Elementary Function81.9Continuity of Functions

§1.9 Continuity of Functions I. Continuity at a Point Introduction II. Continuity on an Interval III. Classification of Discontinuous Points IV. Operations on Continuous Functions V. Continuity of Elementary Function

I. Continuity at a Point01xX1Cx1.IncrementIncrement of independent variable Ax = X, -XiIncrementof dependent variable Ay=f(x,+Ax)-f(x)S1.9Continuity of Functions

§1.9 Continuity of Functions I. Continuity at a Point o x y c o x y c o x y c Increment of dependent variable ( ) ( ) 0 x0 y = f x + x − f 1.Increment Increment of independent variable x = x2 − x1

I. Continuity at a Point00xxx2. DefinitionLet f(x) be defined on an open interval containing xoWe say that f(x) is continuous at x, if lim Ay = 0.Ar->0lim Ay = 0 lim[f(x +△r)- f(x)]= 0Ar-→0Ar->0← lim f(x)= f(x)(let x, + Ax = x)x-→Xo1.9Continuity of FunctionsP

§1.9 Continuity of Functions I. Continuity at a Point o x y c o x y c o x y c ( ) lim 0. ( ) 0 0 0  =  → f x x y f x x x We say that is continuous at if Let be defined onan openinterval containing . 2. Definition lim 0 lim[ ( ) ( )] 0 0 0 0 0 =  + − = → → y f x x f x x x     lim ( ) ( ) 0 0 f x f x x x  = → ) 0 (let x + x = x

I. Continuity at a Pointf(x)is continuousat x, lim f(x)= f(xo)x→>xo1) f(x.)exists;lim f(x) = f(xo)2) lim f(x) exists:x→xox-→x03) lim f(x) = f(x)x→x0Other equivalent definitionsf(x) is continuous at x.V>0,38>0, as x-x<8,s.t.f(x)-f(x)<8台 f(x)= f(x,)= f(x)$1.9Continuity of Functions

§1.9 Continuity of Functions ( ) lim ( ) ( ). 0 0 f x x0 f x f x x x  = → is continuousat 2) lim ( ) ; 0 f x exists x→x 1) ( ) ; f x0 exists 3) lim ( ) ( ) 0 0 f x f x x x = → lim ( ) ( ) 0 0 f x f x x x = → 0 f (x)is continuousat x    0,  0, −   , ( )− ( )   0 x0 as x x s.t. f x f Other equivalent definitions ( ) ( ) ( ) 0 0 x0  f x = f x = f − + I. Continuity at a Point

I. Continuity at a PointV3. Left and right continuityf(x) is left continuous at x.olXoxif f(x )= f(x)Vf(x) is right continuous at x,if f(x)= f(x)0xoxI. Continuity on an Intervalf(x) is continuous on interval [a, b] if f (x) iscontinuous at each point of the open interval (a,b)right continuous at aleft continuousatb.81.9Continuity of Functions

§1.9 Continuity of Functions I. Continuity at a Point 3. Left and right continuity 0 f (x)is left continuous at x ( ) ( ) 0 x0 f x = f − if 0 f (x)is right continuous at x ( ) ( ) 0 x0 f x = f + if o x y x0 o x y x0 II. Continuity on an Interval f (x) is continuous on interval [a, b] if f (x) is ◼ continuous at each point of the open interval (a,b) ◼ right continuous at a ◼ left continuous at b

Il. Continuity on an IntervalExample 1 Show that f(x)= xl is continuous at x = 0.x, x≥0yProoff(x)=-x,x0+?lim f(x)= lim -x = 0x-20x-→0Then limx = 0 = f(0)x-0That is ,f(x) is continuous atx-0.81.9Continuity of Functions

§1.9 Continuity of Functions II. Continuity on an Interval Example 1 Show that f (x) = x is continuousat x = 0. Proof , , 0 , 0 ( )    −   = = x x x x f x x o x y lim ( ) lim 0 0 0 = = → + → + f x x x x lim ( ) lim 0 0 0 = − = → − → − f x x x x lim 0 (0) 0 x f x = = → Then That is , f (x) is continuous at x=0

I. Continuity on an IntervalExample 2 Show that y = cosx is continuous on(-oo,+oo)ProofWe need to prove limcosx = cosc.x->cLet be given.x+c-cosx-cosc=sin22Choose S=8,Then 0<x-c<S implies that cosx-cosc<That is , cosx is continuous on (-oo,+o).$1.90Continuity of Functions

§1.9 Continuity of Functions II. Continuity on an Interval Example 2 Show that is continuous on y = cos x (−,+). Proof lim cos x cos c. x c = → Let be given.  cos x − cos c We need to prove 2 sin 2 2sin x + c x − c = −  x − c   Choose  =  , Then 0  x − c   implies that cos x − cos c   That is , cosx is continuous on (−,+)

Ill. Classification of Discontinuous PointsThat f(x) is continuous at a point requires three things:3) lim f(x) = f(x)1) f(x.) exists;2) lim f(x) exists;X→x0x-→XoIf any one of these three fails, then f is discontinuous atx,$1.9Continuity of Functions

§1.9 Continuity of Functions III. Classification of Discontinuous Points That f (x) is continuous at a point requires three things: 2) lim ( ) ; 0 f x exists x→x 1) ( ) ; f x0 exists 3) lim ( ) ( ) 0 0 f x f x x x = → If any one of these three fails, then f is discontinuous at . 0 x

IL.Classification of Discontinuous Pointsjumpremovableyyyy=f(x)y=f(x)y=f(x)y=f(x)>xX★XX0000(a)(b)(c)(d)yoscillatinginfinite=f(x)=1y=sin10(f)(e)$1.9Continuityof Functions

§1.9 Continuity of Functions III. Classification of Discontinuous Points removable jump infinite oscillating