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