Continuity

A function, f, is continuous at a number, a, if1) f(a) is defined

2) exists

3)

Definition

lim ( )x a

f x

lim ( ) ( )x a

f x f a

A function is continuous if there are no breaks in the graph

Visually

At what points on the graph below is the function discontinuous?

Example:

Where are each of these discontinuous?

1)

2)

3)

4)

Examples:

2 2( )

2

x xf x

x

2

10

( )1 0

xf x x

x

2 2

2( ) 2

1 2

x xx

f x xx

( )f x x

Removable Discontinuities Infinite Discontinuities Jump Discontinuities

Three Types of Discontinuity

Function redefined at a point

Removable Discontinuity

Infinite discontinuity

Jump Discontinuity

A function is continuous from the right if

and continuous from the left if

One-sided Continuity

lim ( ) ( )x a

f x f a

lim ( ) ( )x a

f x f a

A function is continuous on an interval if it is continuous at every number on the interval- if f is defined on only one side of an endpoint, it is continuous from the right or left

Continuous on a Interval

Show that is continuous on the interval [-1,1]

- need to show that it is continuous from the right to -1 and continuous from the left to 1

Example:2( ) 1 1f x x