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,x<0lim f(x)= lim x = 00xx-0x-→>0+?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