sec 2.5: continuity intuitively, any function whose graph can be sketched over its domain in one...
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Sec 2.5: Continuity
Intuitively, any function whose graph can be sketched over its domain in one continuous motion without lifting the pencil is an example of a continuous function.
Continuous Function
defined 1)1()1 f
exist 1)(lim)21
xfx
)(lim)1()31xff
x
exist )(lim)2 xfax
1at cont )( xxf
Sec 2.5: Continuity
Continuity at a Point (interior point)
A function f(x) is continues at a point a if
)(lim )()3 xfafax
defined )()1 af
Continuity Test
)(lim )( xfafax
Example: study the continuity at x = -1
defined )4()1 f
)(lim)24
xfx
)(lim)4()34
xffx
exist )(lim)2 xfax
4at discont )( xxf
Sec 2.5: Continuity
Continuity at a Point (interior point)
A function f(x) is continues at a point a if
)(lim )()3 xfafax
defined )()1 af
Continuity Test
)(lim )( xfafax
Example: study the continuity at x = 4
defined )2()1 f
)(lim)22
xfx
)(lim)2()32
xffx
exist )(lim)2 xfax
2at discont )( xxf
Sec 2.5: Continuity
Continuity at a Point (interior point)
A function f(x) is continues at a point a if
)(lim )()3 xfafax
defined )()1 af
Continuity Test
)(lim )( xfafax
Example: study the continuity at x = 2
defined )2()1 f
)(lim)22xf
x
)(lim)2()32xff
x
exist )(lim)2 xfax
2at discont )( xxf
Sec 2.5: Continuity
Continuity at a Point (interior point)
A function f(x) is continues at a point a if
)(lim )()3 xfafax
defined )()1 af
Continuity Test
)(lim )( xfafax
Example: study the continuity at x = -2
Cont from left at a
Sec 2.5: Continuity
Continuity at a Point (end point)
A function f(x) is continues at an end point a if
)(lim )( xfafax
)(lim )( xfafax
exist )(lim)2 xf
ax
)(lim )()3 xfafax
defined )()1 af
exist )(lim)2 xfax
)(lim )()3 xfafax
defined )()1 af
exist )(lim)2 xfax
)(lim )()3 xfafax
defined )()1 af
Cont from right at a
Cont a
removable discontinuity
jump discontinuity
infinitediscontinuity
Which conditions
Sec 2.5: Continuity
Types of Discontinuities.
Later:oscillating discontinuity:
Sec 2.5: Continuity
Sec 2.5: Continuity
Exam1-102
Sec 2.5: Continuity
Sec 2.5: Continuity
Exam1-122
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
Remark
The inverse function of any continuous one-to-one function is also continuous.
Sec 2.5: Continuity
Inverse Functions and Continuity
This result is suggested from the observation that the graph of the inverse, being the reflection of the graph of ƒ across the line y = x
Sec 2.5: Continuity
))((lim xgfax
))(lim( xgfax
continuous
Sec 2.5: Continuity
Exam1-101
Sec 2.5: Continuity
Sec 2.5: Continuity
Geometrically, IVT says that any horizontal line between ƒ(a) and ƒ(b) will cross the curve at least once over the interval [a, b].
Sec 2.5: Continuity
2) y0 between ƒ(a) and ƒ(b)
1) ƒ(x) cont on [a,b] y0=ƒ(c)
c in [a,b]
The Intermediate Value Theorem
One use of the Intermediate Value Theorem is in locating roots of equations as in the following example.
Sec 2.5: Continuity
Sec 2.5: Continuity
E1 TERM-121
Exam1-101
Sec 2.5: Continuity
Sec 2.5: Continuity