A function f is continuous at a number if . f x f akoblbauermath.weebly.com/uploads/1/3/1/9/13192946/2014_continuity.pdf · A function f is continuous at a number a if . There are

A function f is continuous at a number a if .

There are three types of discontinuity to consider:

Removable discontinuities occur at "holes". They are removable because the function could be redefined at that point, thereby removing the discontinuity.

Ex:

Infinite discontinuities occur where the value of the function is becoming very large (going to infinity or negative infinity). These occur at vertical asymptotes.

Jump discontinuities occur where the function "jumps" from one value to another.

One-sided continuity:

A function f is continuous from the right at a number a if .

and f is continuous from the left at a number a if .

→ A function f is continuous on an interval if it is continuous at every number in the interval. (At an endpoint the continuity from the left or right is considered, as applicable.)

→ If f(x) and g(x) are continuous at a and c is a constant, then the

2 4 3 if 3

( ) 3

2 if 3

x xx

f x x

x

lim ( ) ( )x a

f x f a

lim ( ) ( )x a

f x f a

lim ( ) ( )x a

f x f a

Continuity

→ If f(x) and g(x) are continuous at a and c is a constant, then the following functions are also continuous at a:

→ Any polynomial is continuous everywhere, and any rational function is continuous wherever it is defined.

Ex: On what interval(s) is each function continuous?

100 37

2

2

2

( ) ( ) 2 75

2 17( ) ( )

1

1 1( ) ( )

1 1

a f x x x

x xb f x

x

x xc f x x

x x

The Intermediate Value Theorem:Suppose that f(x) is continuous on the closed interval [a, b] and let N be any number strictly between f(a) and f(b). Then there exists a number cin (a, b) such that f(c) = N.

This may make more sense using a visual explanation:

f (a)

f (b)

a b

( )( ) ( ) ( ) ( ) ( ) if ( ) 0

( )

f xf x g x c f x f x g x g a

g x

f (b)

f (a)

a b