2.3 continuity. definition a function y = f(x) is continuous at an interior point c if a function y...

2.3 Continuity

Definition

• A function y = f(x) is continuous at an interior point c if

• A function y = f(x) is continuous at a left endpoint a or a right endpoint b if

lim ( ) ( )x c

f x f c

lim ( ) ( ) or lim ( ) ( )x a x b

f x f a f x f b

Requirements for continuity at x = c• There are three things required for a function

y = f(x) to be continuous at a point x = c.

1. c must be in the domain of f, ( i.e. f(c) is defined)

2.

3.

If any one of the above fails then the function is discontinous at x = c.

lim ( ) must exist and equal a real numberx c

f x

lim ( ) ( )x c

f x f c

Example: Find the points of discontinuity

Find the points of discontinuity of f(x) = int(x)

Types of discontinuity

1. Removable (also called point or hole discontinuity)

2. Jump discontinuity

3. Infinite discontinuity

4. Oscillating discontinuity

Removable

• The exists but is not equal to f(c)

either because f(c) is undefined or defined elsewhere. I.e. wherever there is a hole. You can “remove” the discontinuity by defining f(c) to be

Examples:

limx c

f x

limx c

f x

Jump discontinuity

• The function is discontinuous at x = c because the limit as x approaches c does not exist since lim ( ) lim ( )

x c x cf x f x

Infinite discontinuity

• Infinite discontinuity occurs at x = c when

• I.e. wherever there is a vertical asymptote

• Example: f(x) = 1/x is discontinuous at

x = 0 since

lim ( ) or lim ( )x c x c

f x f x

0 0lim ( ) and lim ( )x x

f x f x

Oscillating discontinuity

• The limit of f(x) as x approaches c does not exists because the y values oscillate as x approaches c.

• Example: f(x) = sin(1/x) is discontinuous at x = 0 since the y values oscillate between 1 and –1 as x approaches 0

Example 1

• Find and classify the points of discontinuity, if any.

• If removable, how could f(x) be defined to remove the discontinuity?

3

2

7 6( )

9

x xf x

x

Example 2



2

1( )

1f x

x

Example 3



2( )

2

xf x

x

Example 4



( ) 3 cosf x x x

Example 5



2

2 3 x<1( )

x 1

xf x

x

Example 6



2

2 x 2( )

-4x+1 x >2

xf x

x

Continuous functions

• A continuous function is a function that is continuous at every point of its domain.

• For example: y = 1/x is a continuous function because it is continuous at every point in its domain. We say that it is continuous on its domain. It is not, however, continuous on the interval [-1,1] for example.

• y = |x| is a continuous over all reals (its domain) so it is a continuous function

Properties of continuous functions

• If f and g are continuous at x = c, then the following are continuous at x = c.

• f + g

• f – g

• fg

• f/g provided g(c) is not zero

• kf where k is a constant

Discuss the continuity of each function

1.

2. F(x) = x x<1

2 x=1

2x-1 x>1

3. G(x) = int (x)

4. y = x2 + 3

2 1( )

1

xf x

x

Continuity of compositions

• If f is continuous at x = c and g is continuous at f(c) then g(f(x)) is continuous at x = c.

Discuss the continuity of the compositions

1.

2. f(g(x)) if f(x) = and g(x) = x2 + 5

3. f(g(x)) if f(x) = and g(x) = x – 1

2

sin

2

x xy

x

1

6x

1

x

Intermediate Value Theorem

• A function y = f(x) that is continuous on

[a, b] takes on every y-value between f(a) and f(b)

This theorem guarantees that if a function is continuous over an interval then the graph will be connected over the interval – no breaks, jumps or branches (f(x) = 1/x)

2.3 continuity. definition a function y = f(x) is continuous at an interior point c if a function y...

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