Section 1.4: Continuity and One-Sided Limits

t x

ExampleUse the graph of t(x) to determine the intervals on

which the function is continuous.

6, 3 3,0 0,2 2,5

5,

Example 2

Sketch a graph of the function with the following characteristics:

1.Continuous on the interval:

[-7,4)U(4,6)U(6,7) and Range: (-∞,10)

2. exists, Continuous on the

interval: (-∞,-3)U(-3,∞).

limx 3

f x

Example 3

Find values of a and b that makes f(x) continuous.

f x ax 3 if x 5

8 if x 5

x 2 +bx +1 if x 5

When x=5, all three pieces must have a limit of 8.

5a 5

ax 3 8

a 5 3 8

a 1

5b 28 8

x 2 bx 3 8

5 2 b 5 3 8

5b 20

b 4

Continuity at a Point

A function f is continuous at c if the following three conditions are met:

1. is defined.

2. exists.

3.

( )f c

lim ( )x c

f x

lim ( ) ( )x c

f x f c

c

L

f(x)

x

For every question of this type, you need (1), (2), (3), conclusion.

Example 1

Show is continuous at x = 0.

f x 2 1 x 2

1. f 0

2 1 02

1The function is clearly

defined at x= 0

2. limx 0

2 1 x 2

2 1 02

1With direct

substitution the limit clearly exists at x=0

3. f 0 limx 0

2 1 x 2The value of the

function clearly equals the limit at x=0

f is continuous at x = 0

Example 2

Show is not continuous at x = 2. 8 1 if 2

10 if 2

x xf x

x

1. 2f 10The function is clearly

10 at x = 2

2

lim 8 1x

x

8 2 1 15

With direct substitution the limit clearly exists at x=0

2

3. 2 lim x

f f x

The value of the function clearly does not equal the limit at x=2

f is not continuous at x = 2

The behavior as x approaches 2 is dictated by 8x-1

2

2. lim x

f x

DiscontinuityIf f is not continuous at a, we say f is discontinuous

at a, or f has a discontinuity at a.

Typically a hole in the curve

Step/Gap Asymptote

Types Of Discontinuities

RemovableAble to remove the “hole” by

defining f at one point

Non-RemovableNOT able to remove the “hole” by defining f at

one point

Example

Find the x-value(s) at which is not continuous. Which of the discontinuities are removable?

22 153( ) x x

xf x

There is a discontinuity at x=-3 because this makes the

denominator zero.

If f can be reduced, then the discontinuity is removable:

2 5 3

3

x x

xf x

2 5 3

3

x x

x

2 5x This is the

same function as f except at

x=-3

f has a removable discontinuity at x = 0

One-Sided Limits: Left-HandIf f(x) becomes arbitrarily close to a single REAL

number L as x approaches c from values less than c, the left-hand limit is L.

The limit of f(x)…

as x approaches c from the left…

is L.

Notation:

c

L

f(x)

x

lim ( )x c

f x L

lim ( )x c

f x L

One-Sided Limits: Right-HandIf f(x) becomes arbitrarily close to a single REAL

number L as x approaches c from values greater than c, the right-hand limit is L.

The limit of f(x)…

as x approaches c from the right…

is L.

Notation:

c

L

f(x)

x

Example 1

Evaluate the following limits for

f x x

limx 1

f x

limx 1

f x

limx 1

f x

1

0

DNE

3.5

limx

f x

3.5

limx

f x

3.5

limx

f x

3

3

3

Example 2

Sketch a graph of a function with the following characteristics:

4

lim 1x

f x

4

lim 6x

f x

Example 3

Analytically find . 12

24

3 if 4lim if

if 4x

x xf x f x

x x

12If is approaching 4 from the left, the function is defined by 3x x

4

limx

f x

12

4lim 3

xx

1

2 4 3

4

Therefore lim 5x

f x

5

The Existence of a Limit

Let f be a function and let c be real numbers. The limit of f(x) as x approaches c is L if and only if

lim ( ) lim ( )x c x c

f x L f x

c

L

f(x)

x

Example 1Analytically show that .

limx 2

x 2 1 1

Evaluate the right hand limit at 2 :x

limx 2

x 2 1

limx 2

x 2 1

2 2 1

2 1

x

if 2

if 2

x

x

1

Evaluate the left hand limit at 2 :x

limx 2

x 2 1

limx 2

x 2 1

2 2 1

1

Therefore limx 2

x 2 1 1

Use when x>2

Use when x<2

You must use the piecewise equation:

2 +1

2 +1

x

x

Example2Analytically show that is continuous at x = -1. = 1f x x

Evaluate the right hand limit at 1:x

1lim 1

xx

1lim 1

xx

1 1

f x

if 1

if 1

x

x

0

Evaluate the left hand limit at 1:x

1lim 1

xx

1lim 1

xx

1 1 0

Therefore is continuous at 1f x x

Use when x>-1

Use when x<-1

You must use the piecewise equation:

1

1

x

x

1. 1f 1 1 0

1

2. Find limx

f x

0

1

3. 1 lim x

f f x

Continuity on a Closed Interval

A function f is continuous on [a, b] if it is continuous on (a, b) and

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

f x f a and f x f b

a

f(a)

f(b)

x

b

ExampleDiscuss the the continuity of

f x 1 1 x 2

The domain of f is [-1,1]. From our limit properties, we can say it

is continuous on (-1,1)

By direct substitution:

limx 1

f x

1 1 1 2

1

f 1

limx 1

f x

1 1 1 2

1

f 1

Is the middle is continuous?

Are the one-sided limits of the endpoints equal to the functional value?

f is continuous on [-1,1]

Properties of ContinuityIf b is a real number and f and g are continuous at x = c,

then following functions are also continuous at c:

1. Scalar Multiple:

2. Sum/Difference:

3. Product:

3. Quotient: if

4. Composition:

Example: Since are continuous, is continuous too.

f og x

f x g x

f x g x

b f x

f x g x

g c 0

f x 2x and g x x 2

h x 2x x 2

Intermediate Value TheoremIf f is continuous on the closed interval [a, b] and k

is any number between f(a) and f(b), then there is at least one number c in [a, b] such that:

( )f c k

a b

f(a)

f(b)

k

c

ExampleUse the intermediate value theorem to show

has at least one root.

f x 4x 3 6x 2 3x 2

f 0 4 0 3 6 0 2 3 0 2

2

f 2 4 2 3 6 2 2 3 2 2

12

Find an output greater than zero

Find an output less than zero

Since f(0) < 0 and f(2) > 0

There must be some c such that f(0) = 0 by the IVTThe IVT can be used since f

is continuous on [-∞,∞].

ExampleShow that has at least one solution on the

interval .

cos x x 3 x

f 4 cos

4 4 3

4

1.008

f 2 cos

2 2 3

2

2.305

Find an output less than zero

Find an output greater than zero

Since and

There must be some c such that cos(c) = c3 - c by the IVT

The IVT can be used since the left and right side are

both continuous on [-∞,∞].

4 ,

2 Solve the equation for zero.

cos x x 3 x 0

f x

f 4 0

f 2 0