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In this section, we will learn about: Limits and continuity of various types of functions.

1.
14

PARTIAL DERIVATIVES

PARTIAL DERIVATIVES

2.
PARTIAL DERIVATIVES

14.2

Limits and Continuity

In this section, we will learn about:

Limits and continuity of

various types of functions.

14.2

Limits and Continuity

In this section, we will learn about:

Limits and continuity of

various types of functions.

3.
LIMITS AND CONTINUITY

Let’s compare the behavior of the functions

2 2 2 2

sin( x y ) x y

f ( x, y ) 2 2

and g ( x, y ) 2 2

x y x y

as x and y both approach 0

(and thus the point (x, y) approaches

the origin).

Let’s compare the behavior of the functions

2 2 2 2

sin( x y ) x y

f ( x, y ) 2 2

and g ( x, y ) 2 2

x y x y

as x and y both approach 0

(and thus the point (x, y) approaches

the origin).

4.
LIMITS AND CONTINUITY

The following tables show values of f(x, y)

and g(x, y), correct to three decimal places,

for points (x, y) near the origin.

The following tables show values of f(x, y)

and g(x, y), correct to three decimal places,

for points (x, y) near the origin.

5.
LIMITS AND CONTINUITY Table 1

This table shows values of f(x, y).

This table shows values of f(x, y).

6.
LIMITS AND CONTINUITY Table 2

This table shows values of g(x, y).

This table shows values of g(x, y).

7.
LIMITS AND CONTINUITY

Notice that neither function is defined

at the origin.

It appears that, as (x, y) approaches (0, 0),

the values of f(x, y) are approaching 1, whereas

the values of g(x, y) aren’t approaching any

number.

Notice that neither function is defined

at the origin.

It appears that, as (x, y) approaches (0, 0),

the values of f(x, y) are approaching 1, whereas

the values of g(x, y) aren’t approaching any

number.

8.
LIMITS AND CONTINUITY

It turns out that these guesses based on

numerical evidence are correct.

Thus, we write:

sin( x 2 y 2 )

lim 2 2

1

( x , y ) (0,0) x y

2 2

lim

x y

2 2

does not exist.

( x , y ) (0,0) x y

It turns out that these guesses based on

numerical evidence are correct.

Thus, we write:

sin( x 2 y 2 )

lim 2 2

1

( x , y ) (0,0) x y

2 2

lim

x y

2 2

does not exist.

( x , y ) (0,0) x y

9.
LIMITS AND CONTINUITY

In general, we use the notation

lim f ( x, y ) L

( x , y ) ( a ,b )

to indicate that:

The values of f(x, y) approach the number L

as the point (x, y) approaches the point (a, b)

along any path that stays within the domain of f.

In general, we use the notation

lim f ( x, y ) L

( x , y ) ( a ,b )

to indicate that:

The values of f(x, y) approach the number L

as the point (x, y) approaches the point (a, b)

along any path that stays within the domain of f.

10.
LIMITS AND CONTINUITY

In other words, we can make the values

of f(x, y) as close to L as we like by taking

the point (x, y) sufficiently close to the point

(a, b), but not equal to (a, b).

A more precise definition follows.

In other words, we can make the values

of f(x, y) as close to L as we like by taking

the point (x, y) sufficiently close to the point

(a, b), but not equal to (a, b).

A more precise definition follows.

11.
LIMIT OF A FUNCTION Definition 1

Let f be a function of two variables

whose domain D includes points arbitrarily

close to (a, b).

Then, we say that the limit of f(x, y)

as (x, y) approaches (a, b) is L.

Let f be a function of two variables

whose domain D includes points arbitrarily

close to (a, b).

Then, we say that the limit of f(x, y)

as (x, y) approaches (a, b) is L.

12.
LIMIT OF A FUNCTION Definition 1

We write

lim f ( x, y ) L

( x , y ) ( a ,b )

For every number ε > 0, there is a corresponding

number δ > 0 such that,

2 2

if ( x, y ) D and 0 ( x a ) ( y b)

then | f ( x, y ) L |

We write

lim f ( x, y ) L

( x , y ) ( a ,b )

For every number ε > 0, there is a corresponding

number δ > 0 such that,

2 2

if ( x, y ) D and 0 ( x a ) ( y b)

then | f ( x, y ) L |

13.
LIMIT OF A FUNCTION

Other notations for the limit in Definition 1

lim f ( x, y ) L

x a

y b

f ( x, y ) L as ( x, y ) (a, b)

Other notations for the limit in Definition 1

lim f ( x, y ) L

x a

y b

f ( x, y ) L as ( x, y ) (a, b)

14.
LIMIT OF A FUNCTION

Notice that:

| f ( x, y ) L | is the distance between

the numbers f(x, y) and L

( x a) 2 ( y b) 2 is the distance between

the point (x, y) and the point (a, b).

Notice that:

| f ( x, y ) L | is the distance between

the numbers f(x, y) and L

( x a) 2 ( y b) 2 is the distance between

the point (x, y) and the point (a, b).

15.
LIMIT OF A FUNCTION

Thus, Definition 1 says that the distance

between f(x, y) and L can be made arbitrarily

small by making the distance from (x, y) to

(a, b) sufficiently small (but not 0).

Thus, Definition 1 says that the distance

between f(x, y) and L can be made arbitrarily

small by making the distance from (x, y) to

(a, b) sufficiently small (but not 0).

16.
LIMIT OF A FUNCTION

The figure illustrates Definition 1

by means of an arrow diagram.

The figure illustrates Definition 1

by means of an arrow diagram.

17.
LIMIT OF A FUNCTION

If any small interval (L – ε, L + ε) is given around L,

then we can find a disk Dδ with center (a, b) and

radius δ > 0 such that:

f maps all the points in Dδ [except possibly (a, b)]

into the interval (L – ε, L + ε).

If any small interval (L – ε, L + ε) is given around L,

then we can find a disk Dδ with center (a, b) and

radius δ > 0 such that:

f maps all the points in Dδ [except possibly (a, b)]

into the interval (L – ε, L + ε).

18.
LIMIT OF A FUNCTION

Another illustration of

Definition 1

is given here, where the

surface S

is the graph of f.

Another illustration of

Definition 1

is given here, where the

surface S

is the graph of f.

19.
LIMIT OF A FUNCTION

If ε > 0 is given, we can find δ > 0 such that,

if (x, y) is restricted to lie in the disk Dδ and

(x, y) ≠ (a, b), then

The corresponding

part of S lies between

the horizontal planes

z = L – ε and

z = L + ε.

If ε > 0 is given, we can find δ > 0 such that,

if (x, y) is restricted to lie in the disk Dδ and

(x, y) ≠ (a, b), then

The corresponding

part of S lies between

the horizontal planes

z = L – ε and

z = L + ε.

20.
SINGLE VARIABLE FUNCTIONS

For functions of a single variable, when we

let x approach a, there are only two possible

directions of approach, from the left or from

the right.

We recall from Chapter 2 that, if lim f ( x) lim f ( x),

x a x a

then lim f ( x) does not exist.

x a

For functions of a single variable, when we

let x approach a, there are only two possible

directions of approach, from the left or from

the right.

We recall from Chapter 2 that, if lim f ( x) lim f ( x),

x a x a

then lim f ( x) does not exist.

x a

21.
DOUBLE VARIABLE FUNCTIONS

For functions of two

variables, the situation

is not as simple.

For functions of two

variables, the situation

is not as simple.

22.
DOUBLE VARIABLE FUNCTIONS

This is because we can let (x, y) approach

(a, b) from an infinite number of directions

in any manner whatsoever as long as (x, y)

stays within the domain of f.

This is because we can let (x, y) approach

(a, b) from an infinite number of directions

in any manner whatsoever as long as (x, y)

stays within the domain of f.

23.
LIMIT OF A FUNCTION

Definition 1 refers only to the distance

between (x, y) and (a, b).

It does not refer to the direction of approach.

Definition 1 refers only to the distance

between (x, y) and (a, b).

It does not refer to the direction of approach.

24.
LIMIT OF A FUNCTION

Therefore, if the limit exists, then f(x, y)

must approach the same limit no matter

how (x, y) approaches (a, b).

Therefore, if the limit exists, then f(x, y)

must approach the same limit no matter

how (x, y) approaches (a, b).

25.
LIMIT OF A FUNCTION

Thus, if we can find two different paths of

approach along which the function f(x, y)

has different limits, then it follows that

lim f ( x, y ) does not exist.

( x , y ) ( a ,b )

Thus, if we can find two different paths of

approach along which the function f(x, y)

has different limits, then it follows that

lim f ( x, y ) does not exist.

( x , y ) ( a ,b )

26.
LIMIT OF A FUNCTION

f(x, y) → L1 as (x, y) → (a, b) along a path C1

f(x, y) → L2 as (x, y) → (a, b) along a path C2,

where L1 ≠ L2,

then lim f ( x, y )

( x , y ) ( a ,b )

does not exist.

f(x, y) → L1 as (x, y) → (a, b) along a path C1

f(x, y) → L2 as (x, y) → (a, b) along a path C2,

where L1 ≠ L2,

then lim f ( x, y )

( x , y ) ( a ,b )

does not exist.

27.
LIMIT OF A FUNCTION Example 1

Show that 2

x y 2

lim

( x , y ) (0,0) x 2 y 2

does not exist.

Let f(x, y) = (x2 – y2)/(x2 + y2).

Show that 2

x y 2

lim

( x , y ) (0,0) x 2 y 2

does not exist.

Let f(x, y) = (x2 – y2)/(x2 + y2).

28.
LIMIT OF A FUNCTION Example 1

First, let’s approach (0, 0) along

the x-axis.

Then, y = 0 gives f(x, 0) = x2/x2 = 1 for all x ≠ 0.

So, f(x, y) → 1 as (x, y) → (0, 0) along the x-axis.

First, let’s approach (0, 0) along

the x-axis.

Then, y = 0 gives f(x, 0) = x2/x2 = 1 for all x ≠ 0.

So, f(x, y) → 1 as (x, y) → (0, 0) along the x-axis.

29.
LIMIT OF A FUNCTION Example 1

We now approach along the y-axis by

putting x = 0.

Then, f(0, y) = –y2/y2 = –1 for all y ≠ 0.

So, f(x, y) → –1 as (x, y) → (0, 0) along the y-axis.

We now approach along the y-axis by

putting x = 0.

Then, f(0, y) = –y2/y2 = –1 for all y ≠ 0.

So, f(x, y) → –1 as (x, y) → (0, 0) along the y-axis.

30.
LIMIT OF A FUNCTION Example 1

Since f has two different limits along

two different lines, the given limit does

not exist.

This confirms

the conjecture we

made on the basis

of numerical evidence

at the beginning

of the section.

Since f has two different limits along

two different lines, the given limit does

not exist.

This confirms

the conjecture we

made on the basis

of numerical evidence

at the beginning

of the section.

31.
LIMIT OF A FUNCTION Example 2

If xy

f ( x, y ) 2 2

x y

lim f ( x, y )

( x , y ) (0,0)

If xy

f ( x, y ) 2 2

x y

lim f ( x, y )

( x , y ) (0,0)

32.
LIMIT OF A FUNCTION Example 2

If y = 0, then f(x, 0) = 0/x2 = 0.

Therefore,

f(x, y) → 0 as (x, y) → (0, 0) along the x-axis.

If y = 0, then f(x, 0) = 0/x2 = 0.

Therefore,

f(x, y) → 0 as (x, y) → (0, 0) along the x-axis.

33.
LIMIT OF A FUNCTION Example 2

If x = 0, then f(0, y) = 0/y2 = 0.

So,

f(x, y) → 0 as (x, y) → (0, 0) along the y-axis.

If x = 0, then f(0, y) = 0/y2 = 0.

So,

f(x, y) → 0 as (x, y) → (0, 0) along the y-axis.

34.
LIMIT OF A FUNCTION Example 2

Although we have obtained identical limits

along the axes, that does not show that

the given limit is 0.

Although we have obtained identical limits

along the axes, that does not show that

the given limit is 0.

35.
LIMIT OF A FUNCTION Example 2

Let’s now approach (0, 0) along another

line, say y = x.

For all x ≠ 0,

x2 1

f ( x, x ) 2 2

x x 2

Therefore,

f ( x, y ) 1

2 as ( x, y ) (0, 0) along y x

Let’s now approach (0, 0) along another

line, say y = x.

For all x ≠ 0,

x2 1

f ( x, x ) 2 2

x x 2

Therefore,

f ( x, y ) 1

2 as ( x, y ) (0, 0) along y x

36.
LIMIT OF A FUNCTION Example 2

Since we have obtained different limits

along different paths, the given limit does

not exist.

Since we have obtained different limits

along different paths, the given limit does

not exist.

37.
LIMIT OF A FUNCTION

This figure sheds

some light on

Example 2.

The ridge that occurs

above the line y = x

corresponds to the fact

that f(x, y) = ½ for all

points (x, y) on that line

except the origin.

This figure sheds

some light on

Example 2.

The ridge that occurs

above the line y = x

corresponds to the fact

that f(x, y) = ½ for all

points (x, y) on that line

except the origin.

38.
LIMIT OF A FUNCTION Example 3

If xy 2

f ( x, y ) 2 4

x y

lim f ( x, y )

( x , y ) (0,0)

If xy 2

f ( x, y ) 2 4

x y

lim f ( x, y )

( x , y ) (0,0)

39.
LIMIT OF A FUNCTION Example 3

With the solution of Example 2 in mind,

let’s try to save time by letting (x, y) → (0, 0)

along any nonvertical line through the origin.

With the solution of Example 2 in mind,

let’s try to save time by letting (x, y) → (0, 0)

along any nonvertical line through the origin.

40.
LIMIT OF A FUNCTION Example 3

Then, y = mx, where m is the slope,

and f ( x, y ) f ( x, mx)

x(mx) 2

2 4

x (mx)

m2 x3

2 4 4

x m x

m2 x

4 2

1 m x

Then, y = mx, where m is the slope,

and f ( x, y ) f ( x, mx)

x(mx) 2

2 4

x (mx)

m2 x3

2 4 4

x m x

m2 x

4 2

1 m x

41.
LIMIT OF A FUNCTION Example 3

f(x, y) → 0 as (x, y) → (0, 0) along y = mx

Thus, f has the same limiting value along

every nonvertical line through the origin.

f(x, y) → 0 as (x, y) → (0, 0) along y = mx

Thus, f has the same limiting value along

every nonvertical line through the origin.

42.
LIMIT OF A FUNCTION Example 3

However, that does not show that

the given limit is 0.

This is because, if we now let

(x, y) → (0, 0) along the parabola x = y2

we have: 2 2 4

2 y y y 1

f ( x, y ) f ( y , y ) 2 2 4

4

(y ) y 2y 2

So,

f(x, y) → ½ as (x, y) → (0, 0) along x = y2

However, that does not show that

the given limit is 0.

This is because, if we now let

(x, y) → (0, 0) along the parabola x = y2

we have: 2 2 4

2 y y y 1

f ( x, y ) f ( y , y ) 2 2 4

4

(y ) y 2y 2

So,

f(x, y) → ½ as (x, y) → (0, 0) along x = y2

43.
LIMIT OF A FUNCTION Example 3

Since different paths lead to different

limiting values, the given limit does not

Since different paths lead to different

limiting values, the given limit does not

44.
LIMIT OF A FUNCTION

Now, let’s look at limits

that do exist.

Now, let’s look at limits

that do exist.

45.
LIMIT OF A FUNCTION

Just as for functions of one variable,

the calculation of limits for functions of

two variables can be greatly simplified

by the use of properties of limits.

Just as for functions of one variable,

the calculation of limits for functions of

two variables can be greatly simplified

by the use of properties of limits.

46.
LIMIT OF A FUNCTION

The Limit Laws listed in Section 2.3 can be

extended to functions of two variables.

For instance,

The limit of a sum is the sum of the limits.

The limit of a product is the product of the limits.

The Limit Laws listed in Section 2.3 can be

extended to functions of two variables.

For instance,

The limit of a sum is the sum of the limits.

The limit of a product is the product of the limits.

47.
LIMIT OF A FUNCTION Equations 2

In particular, the following equations

are true.

lim x a

( x , y ) ( a ,b )

lim y b

( x , y ) ( a ,b )

lim c c

( x , y ) ( a ,b )

In particular, the following equations

are true.

lim x a

( x , y ) ( a ,b )

lim y b

( x , y ) ( a ,b )

lim c c

( x , y ) ( a ,b )

48.
LIMIT OF A FUNCTION Equations 2

The Squeeze Theorem

also holds.

The Squeeze Theorem

also holds.

49.
LIMIT OF A FUNCTION Example 4

Find 2

3x y

lim

( x , y ) (0,0) x 2 y 2

if it exists.

Find 2

3x y

lim

( x , y ) (0,0) x 2 y 2

if it exists.

50.
LIMIT OF A FUNCTION Example 4

As in Example 3, we could show that

the limit along any line through the origin

is 0.

However, this doesn’t prove that

the given limit is 0.

As in Example 3, we could show that

the limit along any line through the origin

is 0.

However, this doesn’t prove that

the given limit is 0.

51.
LIMIT OF A FUNCTION Example 4

However, the limits along the parabolas

y = x2 and x = y2 also turn out to be 0.

So, we begin to suspect that the limit

does exist and is equal to 0.

However, the limits along the parabolas

y = x2 and x = y2 also turn out to be 0.

So, we begin to suspect that the limit

does exist and is equal to 0.

52.
LIMIT OF A FUNCTION Example 4

Let ε > 0.

We want to find δ > 0 such that

2

2 2 3x y

if 0 x y then 2 2

0

x y

2

2 2 3x | y |

that is, if 0 x y then 2 2

x y

Let ε > 0.

We want to find δ > 0 such that

2

2 2 3x y

if 0 x y then 2 2

0

x y

2

2 2 3x | y |

that is, if 0 x y then 2 2

x y

53.
LIMIT OF A FUNCTION Example 4

x2 ≤ x2 = y2 since y2 ≥ 0

Thus,

x2/(x2 + y2) ≤ 1

x2 ≤ x2 = y2 since y2 ≥ 0

Thus,

x2/(x2 + y2) ≤ 1

54.
LIMIT OF A FUNCTION E. g. 4—Equation 3

2

3x | y | 2 2 2

2 2

3 | y | 3 y 3 x y

x y

2

3x | y | 2 2 2

2 2

3 | y | 3 y 3 x y

x y

55.
LIMIT OF A FUNCTION Example 4

Thus, if we choose δ = ε/3

2 2

and let 0 x y

3x 2 y 2 2

2 2

0 3 x y 3 3

x y 3

Thus, if we choose δ = ε/3

2 2

and let 0 x y

3x 2 y 2 2

2 2

0 3 x y 3 3

x y 3

56.
LIMIT OF A FUNCTION Example 4

Hence, by Definition 1,

2

3x y

lim 0

( x , y ) (0,0) x 2 y 2

Hence, by Definition 1,

2

3x y

lim 0

( x , y ) (0,0) x 2 y 2

57.
CONTINUITY OF SINGLE VARIABLE FUNCTIONS

Recall that evaluating limits of continuous

functions of a single variable is easy.

It can be accomplished by direct substitution.

This is because the defining property of

a continuous function is lim f ( x) f ( a )

x a

Recall that evaluating limits of continuous

functions of a single variable is easy.

It can be accomplished by direct substitution.

This is because the defining property of

a continuous function is lim f ( x) f ( a )

x a

58.
CONTINUITY OF DOUBLE VARIABLE FUNCTIONS

Continuous functions of two variables

are also defined by the direct substitution

Continuous functions of two variables

are also defined by the direct substitution

59.
CONTINUITY Definition 4

A function f of two variables is called

continuous at (a, b) if

lim f ( x, y ) f ( a , b )

( x , y ) ( a ,b )

We say f is continuous on D if f is

continuous at every point (a, b) in D.

A function f of two variables is called

continuous at (a, b) if

lim f ( x, y ) f ( a , b )

( x , y ) ( a ,b )

We say f is continuous on D if f is

continuous at every point (a, b) in D.

60.
The intuitive meaning of continuity is that,

if the point (x, y) changes by a small amount,

then the value of f(x, y) changes by a small

This means that a surface that is the graph of

a continuous function has no hole or break.

if the point (x, y) changes by a small amount,

then the value of f(x, y) changes by a small

This means that a surface that is the graph of

a continuous function has no hole or break.

61.
Using the properties of limits, you can see

that sums, differences, products, quotients

of continuous functions are continuous on

their domains.

Let’s use this fact to give examples

of continuous functions.

that sums, differences, products, quotients

of continuous functions are continuous on

their domains.

Let’s use this fact to give examples

of continuous functions.

62.
A polynomial function of two variables

(polynomial, for short) is a sum of terms

of the form cxmyn,

c is a constant.

m and n are nonnegative integers.

(polynomial, for short) is a sum of terms

of the form cxmyn,

c is a constant.

m and n are nonnegative integers.

63.
RATIONAL FUNCTION

A rational function is

a ratio of polynomials.

A rational function is

a ratio of polynomials.

64.
RATIONAL FUNCTION VS. POLYNOMIAL

4 3 2 4

f ( x, y ) x 5 x y 6 xy 7 y 6

is a polynomial.

2 xy 1

g ( x, y ) 2 2

x y

is a rational function.

4 3 2 4

f ( x, y ) x 5 x y 6 xy 7 y 6

is a polynomial.

2 xy 1

g ( x, y ) 2 2

x y

is a rational function.

65.
The limits in Equations 2 show that

the functions

f(x, y) = x, g(x, y) = y, h(x, y) = c

are continuous.

the functions

f(x, y) = x, g(x, y) = y, h(x, y) = c

are continuous.

66.
CONTINUOUS POLYNOMIALS

Any polynomial can be built up out

of the simple functions f, g, and h

by multiplication and addition.

It follows that all polynomials are continuous

on R2.

Any polynomial can be built up out

of the simple functions f, g, and h

by multiplication and addition.

It follows that all polynomials are continuous

on R2.

67.
CONTINUOUS RATIONAL FUNCTIONS

Likewise, any rational function is

continuous on its domain because it is

a quotient of continuous functions.

Likewise, any rational function is

continuous on its domain because it is

a quotient of continuous functions.

68.
CONTINUITY Example 5

2 3 3 2

lim ( x y x y 3 x 2 y )

( x , y ) (1,2)

2 3 3 2

f ( x, y ) x y x y 3x 2 y is a polynomial.

Thus, it is continuous everywhere.

2 3 3 2

lim ( x y x y 3 x 2 y )

( x , y ) (1,2)

2 3 3 2

f ( x, y ) x y x y 3x 2 y is a polynomial.

Thus, it is continuous everywhere.

69.
CONTINUITY Example 5

Hence, we can find the limit by direct

substitution:

2 3 3 2

lim ( x y x y 3x 2 y )

( x , y ) (1,2)

2 3 3 2

1 2 1 2 3 1 2 2

11

Hence, we can find the limit by direct

substitution:

2 3 3 2

lim ( x y x y 3x 2 y )

( x , y ) (1,2)

2 3 3 2

1 2 1 2 3 1 2 2

11

70.
CONTINUITY Example 6

Where is the function

2 2

x y

f ( x, y ) 2 2

x y

Where is the function

2 2

x y

f ( x, y ) 2 2

x y

71.
CONTINUITY Example 6

The function f is discontinuous at (0, 0)

because it is not defined there.

Since f is a rational function, it is continuous

on its domain, which is the set

D = {(x, y) | (x, y) ≠ (0, 0)}

The function f is discontinuous at (0, 0)

because it is not defined there.

Since f is a rational function, it is continuous

on its domain, which is the set

D = {(x, y) | (x, y) ≠ (0, 0)}

72.
CONTINUITY Example 7

2 2

Let x y

2 2

if ( x, y ) (0, 0)

g ( x, y ) x y

0 if ( x, y ) (0, 0)

Here, g is defined at (0, 0).

However, it is still discontinuous there because

lim g ( x, y )

( x , y ) (0,0)

does not exist (see Example 1).

2 2

Let x y

2 2

if ( x, y ) (0, 0)

g ( x, y ) x y

0 if ( x, y ) (0, 0)

Here, g is defined at (0, 0).

However, it is still discontinuous there because

lim g ( x, y )

( x , y ) (0,0)

does not exist (see Example 1).

73.
CONTINUITY Example 8

2

3x y

2 2

if ( x, y ) (0, 0)

f ( x, y ) x y

0 if ( x, y ) (0, 0)

2

3x y

2 2

if ( x, y ) (0, 0)

f ( x, y ) x y

0 if ( x, y ) (0, 0)

74.
CONTINUITY Example 8

We know f is continuous for (x, y) ≠ (0, 0)

since it is equal to a rational function there.

Also, from Example 4, we have:

2

3x y

lim f ( x, y ) lim

( x , y ) (0,0) ( x , y ) (0,0) x 2 y 2

0 f (0, 0)

We know f is continuous for (x, y) ≠ (0, 0)

since it is equal to a rational function there.

Also, from Example 4, we have:

2

3x y

lim f ( x, y ) lim

( x , y ) (0,0) ( x , y ) (0,0) x 2 y 2

0 f (0, 0)

75.
CONTINUITY Example 8

Thus, f is continuous at (0, 0).

So, it is continuous on R2.

Thus, f is continuous at (0, 0).

So, it is continuous on R2.

76.
This figure shows the

graph of

the continuous function

in Example 8.

graph of

the continuous function

in Example 8.

77.
COMPOSITE FUNCTIONS

Just as for functions of one variable,

composition is another way of combining

two continuous functions to get a third.

Just as for functions of one variable,

composition is another way of combining

two continuous functions to get a third.

78.
COMPOSITE FUNCTIONS

In fact, it can be shown that, if f is

a continuous function of two variables and

g is a continuous function of a single variable

defined on the range of f, then

The composite function h = g ◦ f defined by

h(x, y) = g(f(x, y)) is also a continuous function.

In fact, it can be shown that, if f is

a continuous function of two variables and

g is a continuous function of a single variable

defined on the range of f, then

The composite function h = g ◦ f defined by

h(x, y) = g(f(x, y)) is also a continuous function.

79.
COMPOSITE FUNCTIONS Example 9

Where is the function h(x, y) = arctan(y/x)

The function f(x, y) = y/x is a rational function

and therefore continuous except on the line x = 0.

The function g(t) = arctan t is continuous

everywhere.

Where is the function h(x, y) = arctan(y/x)

The function f(x, y) = y/x is a rational function

and therefore continuous except on the line x = 0.

The function g(t) = arctan t is continuous

everywhere.

80.
COMPOSITE FUNCTIONS Example 9

So, the composite function

g(f(x, y)) = arctan(y, x) = h(x, y)

is continuous except where x = 0.

So, the composite function

g(f(x, y)) = arctan(y, x) = h(x, y)

is continuous except where x = 0.

81.
COMPOSITE FUNCTIONS Example 9

The figure shows the

break in the graph

of h above the y-axis.

The figure shows the

break in the graph

of h above the y-axis.

82.
FUNCTIONS OF THREE OR MORE VARIABLES

Everything that we have done in

this section can be extended to functions

of three or more variables.

Everything that we have done in

this section can be extended to functions

of three or more variables.

83.
MULTIPLE VARIABLE FUNCTIONS

The notation

lim f ( x, y , z ) L

( x , y , z ) ( a ,b , c )

means that:

The values of f(x, y, z) approach the number L

as the point (x, y, z) approaches the point (a, b, c)

along any path in the domain of f.

The notation

lim f ( x, y , z ) L

( x , y , z ) ( a ,b , c )

means that:

The values of f(x, y, z) approach the number L

as the point (x, y, z) approaches the point (a, b, c)

along any path in the domain of f.

84.
MULTIPLE VARIABLE FUNCTIONS

The distance between two points (x, y, z)

and (a, b, c) in R3 is given by:

2 2 2

( x a ) ( y b) ( z c )

Thus, we can write the precise definition

as follows.

The distance between two points (x, y, z)

and (a, b, c) in R3 is given by:

2 2 2

( x a ) ( y b) ( z c )

Thus, we can write the precise definition

as follows.

85.
MULTIPLE VARIABLE FUNCTIONS

For every number ε > 0, there is

a corresponding number δ > 0 such that,

if (x, y, z) is in the domain of f

2 2 2

and 0 ( x a) ( y b) ( z c)

|f(x, y, z) – L| < ε

For every number ε > 0, there is

a corresponding number δ > 0 such that,

if (x, y, z) is in the domain of f

2 2 2

and 0 ( x a) ( y b) ( z c)

|f(x, y, z) – L| < ε

86.
MULTIPLE VARIABLE FUNCTIONS

The function f is continuous at (a, b, c)

lim f ( x, y, z ) f (a, b, c)

( x , y , z ) ( a ,b , c )

The function f is continuous at (a, b, c)

lim f ( x, y, z ) f (a, b, c)

( x , y , z ) ( a ,b , c )

87.
MULTIPLE VARIABLE FUNCTIONS

For instance, the function

1

f ( x, y , z ) 2 2 2

x y z 1

is a rational function of three variables.

So, it is continuous at every point in R3

except where x2 + y2 + z2 = 1.

For instance, the function

1

f ( x, y , z ) 2 2 2

x y z 1

is a rational function of three variables.

So, it is continuous at every point in R3

except where x2 + y2 + z2 = 1.

88.
MULTIPLE VARIABLE FUNCTIONS

In other words, it is discontinuous

on the sphere with center the origin

and radius 1.

In other words, it is discontinuous

on the sphere with center the origin

and radius 1.

89.
MULTIPLE VARIABLE FUNCTIONS

If we use the vector notation introduced at

the end of Section 10.1, then we can write

the definitions of a limit for functions of two or

three variables in a single compact form as

If we use the vector notation introduced at

the end of Section 10.1, then we can write

the definitions of a limit for functions of two or

three variables in a single compact form as

90.
MULTIPLE VARIABLE FUNCTIONS Equation 5

If f is defined on a subset D of Rn,

then lim f (x) L means that, for every

x a

number ε > 0, there is a corresponding

number δ > 0 such that

if x D and 0 | x a |

then | f ( x) L |

If f is defined on a subset D of Rn,

then lim f (x) L means that, for every

x a

number ε > 0, there is a corresponding

number δ > 0 such that

if x D and 0 | x a |

then | f ( x) L |

91.
MULTIPLE VARIABLE FUNCTIONS

If n = 1, then

x=x and a=a

So, Equation 5 is just the definition

of a limit for functions of a single variable.

If n = 1, then

x=x and a=a

So, Equation 5 is just the definition

of a limit for functions of a single variable.

92.
MULTIPLE VARIABLE FUNCTIONS

If n = 2, we have

x =

a =

2 2

| x a | ( x a ) ( y b)

So, Equation 5 becomes Definition 1.

If n = 2, we have

x =

a =

2 2

| x a | ( x a ) ( y b)

So, Equation 5 becomes Definition 1.

93.
MULTIPLE VARIABLE FUNCTIONS

If n = 3, then

x = and a =

So, Equation 5 becomes the definition of

a limit of a function of three variables.

If n = 3, then

x =

So, Equation 5 becomes the definition of

a limit of a function of three variables.

94.
MULTIPLE VARIABLE FUNCTIONS

In each case, the definition of continuity

can be written as:

lim f (x) f (a)

x a

In each case, the definition of continuity

can be written as:

lim f (x) f (a)

x a