continuity when will it end. for functions that are "normal" enough, we know immediately...
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Continuity When Will It End
For functions that are "normal" enough, we know immediately whether or not they are continuous at a given point. Nevertheless, the continuity of a function is such an important property that we need a precise definition of continuity at a point: A function f (x) is continuous at c if and only if lim ( ) ( )
x cf x f c
In other words if as you get closer to c from both sides there is a value you are getting closer to and it is the value of the function at c.
Or
If when you trace over a point and you don’t have to lift your pencil to draw the graph, the graph is continuous at that point
1. If f is continuous at every real number c, then f is said to be continuous.
2. If f is not continuous at c, then f is said to be discontinuous at c. The function f can be discontinuous for two distinct reasons: a) f(x) does not have a limit as x approaches c
0
1lim . . .x
D N Ex
The function is discontinuous at x=0
The function is discontinuous at x=1
b) The value of the function at c does not equal the limit or the function is undefined at that value
1
(1) 2
lim ( ) 1x
f
f x
The function is discontinuous at x=1
Endpoint: A function y=f(x) is continuous at a left endpoint or is continuous at a right endpoint b of its domain if
lim ( ) ( ) lim ( ) ( ) .x a x a
f x f a or f x f b respectively
3lim ( ) 2
(3) 2x
f x
f
The graph is continuous at x = 3
3lim ( ) 3
( 3)x
f x
f undefined
The graph is not continuous at x = -3
There are different types of discontinuity
Jump Discontinuity
Infinite DiscontinuityOscillating
Discontinuity
Removable Discontinuity
Removable Discontinuity
In both examples the function would be continuous if the point (1,-1) wasn’t removed
Jump Discontinuity
Jump discontinuity is when the left and right handed limits have different values
Infinite Discontinuity
Infinite discontinuity happens when there is a vertical asymptote.
This is the equation of y=sin(1/x) it oscillates to much to have a limit as therefore it has oscillating discontinuity at x=0
Oscillating Discontinuity
0x
Find the points and the type of discontinuity
3( )
5
xf x
x
2 4( )
2
xg x
x
2 , 1
( ) 2 , 1 3
9, 3
x x
p x x x
x x
Look for asymptotes or holes, in other words values the function is undefined.
In piecewise functions look for where the graph jumps or there are holes.
3( )
5
xf x
x
This graph has infinite discontinuity at x=5.
This graph has a hole, therefore at x= -2 there is removable discontinuity.
2 4 ( 2)( 2)
2 2
x x x
x x
2 4( )
2
xg x
x
2 , 1
( ) 2 , 1 3
9, 3
x x
p x x x
x x
The graph has jump discontinuity at x=1 and removable discontinuity at x=3
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