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