Continuity
Definition of Continuity of a Function of Two Variables
A function of two variables is continuous at a point (a,b)
in an open region R if
lim
( x , y )( a ,b )
f ( x, y)  f (a, b).
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 amount.
This means that a surface that is the graph of
a continuous function has no hole or break.
Using the properties of limits, we can see that
sums,
differences,
products, and
Quotients
composition
of continuous functions are continuous
on their domains.
Example-1
• Evaluate

x 2 y 3  x 3 y 2  3x  2 y .
 x ,y  1, 2 
lim
Example -2. Find the limit and discuss the continuity of
the function
x
lim
( x , y ) (1, 2 ) 2 x  y
Try Me:
Find the limit and discuss the continuity of the function.
x
lim
( x , y ) (1, 2 )
2x  y
Solution
x
1
1
1
lim



( x , y )(1, 2 )
2x  y
2(1)  2
4 2
The function will be continuous when 2x+y > 0.
Why not Me?
• Discuss the continuity of
2
3
x
y

if x , y   0 ,0 
 2
2
f x , y    x  y
 0 if x , y   0 ,0 
Example 3. Use your calculator to fill in the values of the
table below. The first table approaches (0,0) along the line
y=x. The second table approaches (0,0) along the line x=0.
Use the patterns to determine the limit and discuss the
continuity of the function.
z
1
ln( x 2  y 2 )
( x , y ) ( 0 , 0 ) 2
lim
(x,y)
z
(x,y)
(2,2)
-1.04
(0,2)
-0.69
(1,1)
-0.35
(0,1)
0
(.5,.5)
0.35
(0,.5)
0.69
(.1,.1)
1.96
(0,.1)
2.30
(.01,.01) 4.26
z
1
ln( x 2  y 2 )
2
z
(0,.01) 4.61
y
x
Solution: from the graph, it appears that z values get
larger and larger as (x,y) approaches (0,0). Conceptually,
we would expect values of the natural log function to
approach infinity as the inputs approach 0.
conclusion:
z
1
lim
ln( x 2  y 2 )  
( x , y ) ( 0 , 0 ) 2
(x,y)
z
(x,y)
(2,2)
-1.04
(0,2)
-0.69
(1,1)
-0.35
(0,1)
0
(.5,.5)
0.35
(0,.5)
0.69
(.1,.1)
1.96
(0,.1)
2.30
(.01,.01) 4.26
z
1
ln( x 2  y 2 )
2
z
(0,.01) 4.61
y
x
More Variables
• can be extended to functions of three
or more variables.
• The function f is cont. at (a,b,c) iff
lim
 x ,y ,z  a ,b ,c 
f x , y , z   f  a , b , c 
Example-4
• Where does the following function is
continuous?
1
f x , y , z   2
2
2
x  y  z 1
• Solution:
• is continuous at every point in space except
where x2 + y2 + z2 = 1.
• In other words, it is discontinuous on the
sphere with center the origin and radius 1.
True/False