sec 2.5: continuity. study continuity at x = 4 sec 2.5: continuity study continuity at x = 2

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Sec 2.5: CONTINUITY define 1 ) 1 ( ) 1 f exis 1 ) ( lim ) 2 1 x f x ) ( lim ) 1 ( ) 3 1 x f f x ) ( lim ) ( ) 3 exist ) ( lim ) 2 defined ) ( ) 1 conditions 3 x f a f x f a f a x a x 1 at cont ) ( x x f

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Page 1: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

defined 1)1()1 f

exist 1)(lim)21

xfx

)(lim)1()31xff

x

)(lim )()3

exist )(lim)2

defined )()1

conditions 3

xfaf

xf

af

ax

ax

1at cont )( xxf

Page 2: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

defined )4()1 f

exist 4)(lim)24

xfx

)(lim )()3

exist )(lim)2

defined )()1

conditions 3

xfaf

xf

af

ax

ax

4at discont )( xxf

Study continuity at x = 4

Page 3: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

defined )2()1 f

existnot )(lim)22

xfx

)(lim )()3

exist )(lim)2

defined )()1

conditions 3

xfaf

xf

af

ax

ax

2at discont )( xxf

Study continuity at x = 2

Page 4: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

defined )2()1 f

existnot )(lim)22

xfx

)(lim )()3

exist )(lim)2

defined )()1

conditions 3

xfaf

xf

af

ax

ax

2at discont )( xxf

Study continuity at x = 2

Page 5: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

defined )2()1 f

exist )(lim)22xf

x

)(lim )()3

exist )(lim)2

defined )()1

conditions 3

xfaf

xf

af

ax

ax

2at discont )( xxf

Study continuity at x = -2

)(lim)2()32xff

x

Page 6: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

)(lim )()3

exist )(lim)2

defined )()1

xfaf

xf

af

ax

ax

Cont at a

)(lim )()3

exist )(lim)2

defined )()1

xfaf

xf

af

ax

ax

Cont from right at a

)(lim )()3

exist )(lim)2

defined )()1

xfaf

xf

af

ax

ax

Cont from left at a

Page 7: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Three Types of Discontinuities

removable discontinuity

jump discontinuity

infinitediscontinuity

Which condition

s

Page 8: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Three Types of Discontinuities

Page 9: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

Continuouson [a, b]

bf

af

baxf

at left from contiuous )3

at right from contiuous )2

),(every at contiuous )1

Page 10: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

),(on continuous are cos)( ,sin)( Rxxgxxf

The inverse function of any continuous one-to-one function is also continuous.

Page 11: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

Page 12: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

))((lim xgfax

))(lim( xgfax

continuous

Page 13: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

Page 14: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

Sec 2.5: CONTINUITY

Page 15: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

One use of the Intermediate Value Theorem is in locating roots of equations as in the following example.

Sec 2.5: CONTINUITY

Page 16: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

H

R

H

H

H

R

Page 17: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

HR

H

R

H

H

Page 18: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2

H

H

R

H

R

Page 19: Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2