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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
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
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
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
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
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
Three Types of Discontinuities
removable discontinuity
jump discontinuity
infinitediscontinuity
Which condition
s
Three Types of Discontinuities
Sec 2.5: CONTINUITY
Continuouson [a, b]
bf
af
baxf
at left from contiuous )3
at right from contiuous )2
),(every at contiuous )1
Sec 2.5: CONTINUITY
),(on continuous are cos)( ,sin)( Rxxgxxf
The inverse function of any continuous one-to-one function is also continuous.
Sec 2.5: CONTINUITY
Sec 2.5: CONTINUITY
))((lim xgfax
))(lim( xgfax
continuous
Sec 2.5: CONTINUITY
Sec 2.5: CONTINUITY
One use of the Intermediate Value Theorem is in locating roots of equations as in the following example.
Sec 2.5: CONTINUITY
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