Functions of Several
Variables
Faruk Uygul
University of Alberta
Fall 2025
Let us begin by visiting some standard definitions.
A point in R is determined by an ordered triple (a, b, c).
The points ( , , ), ( , → , ) and (→ , , → ).
Figure: The points ( , , ), ( , → , ) and (→ , , → ).
- We use the "right hand rule" to construct the
z-axis
- There are 3-coordinate planes: xy -plane, yz-plane
and xz-plane
- There are 8 octants.
↭ In the first octant, we have: x, y , z ↑ .
We can project a point P(a, b, c) onto the three
coordinate planes.
Definition. We define
R = R ↓ R ↓ R := {(x, y , z) : x, y , z ↔ R}.
Example Sketch the following surfaces in R .
-
z = , y =→
and x = → .
(a) z =
(b) y = →
(c) x = →
Example Sketch the following surfaces in R .
x +y =
and z = .
(a)
x +y =
x +y =
(b) z =
R
(c)
x +y =
and z =
R
A function of two variables (z = f (x, y )) is a rule that
assigns each ordered pair of real numbers (x, y ) in a
set D ↗ R to a unique real number f (x, y ) ↔ R. In this
setting:
D is called the domain of the function The set of all
possible values of f (x, y ) is called the range of the
function.
For the function z = f (x, y ), x and y are called the
independent variables and z is called the dependent
variable.
Similarly we can define functions with three variables
w = f (x, y , z) or even functions with n-variables
z = f (x , x , . . . , xn ).
If z = f (x, y ), then we define the graph of f by the set
Gf := {(x, y , f (x, y )) ↔ R : (x, y ) ↔ Dom(f )}
Note that the graph of
y = f (x) is in R , two-dimensional space
z = f (x, y ) is in R , three-dimensional space
z = f (x , x , . . . , xn ) Rn+ , in the (n + )-dimensional space
Example 1. Consider the function
z = f (x, y ) = sin( x + y ).
↘ Find and sketch the domain of f
↘ Find the range of f
Example 2. Consider the
! function
!
→
f (x, y ) = x → tan ( y ) + x + y → →
↘ Find and sketch the domain of f
→x →y .
Example 3. Consider the
≃ function
f (x, y ) = ln( x + y → ) →
→x .
↘ Find and sketch the domain of f
Sketching the graph of a function z = f (x, y ) can be a
tedious job.
For example, the graphs of z = sin(x) + cos( y ) and
z = (x + y ) →x →y are shown below.
In order to sketch the graph of z = f (x, y ) we make use
of level curves.
Definition The level curves of a function z = f (x, y )
are the curves with equations f (x, y ) = k, where k is a
constant (k ↔ Range(f )). A family of level curves
obtained by different values of k is called a contour
map of f .
Example 4. The graph of z = x + y and its level curves
are as follows.
!
Consider z = f (x, y ) =
→ x → y . Find the domain and
the range of f . Draw a contour map of f by sketching the level curves of
the function for the values k = , , , , . Finally, sketch the graph of f .
Example 5.
Example 6.
Consider z = f (x, y ) = + x + y . Find the domain and
the range of f . Draw a contour map of f by sketching several level curves
of the function. Finally, sketch the graph of f .
Note that in case of a function of 3 variables
w = f (x, y , z):
f (x, y , z) = k
(
k ↔ Ran(f ))
we obtain level surfaces.
Example 7.
Find the domain and range of w = f (x, y , z) = x + y + z .
What are the level surfaces of it?
Limits and Continuity
Faruk Uygul
University of Alberta
Fall 2025
Limits
We now study the continuity of functions with two
variables. Consider a function f (x, y ) and (a, b) → R . We
say that the limit of f at (a, b) is some number L if the
values of f (x, y ) gets arbitrarily close to L as (x, y )
approaches (a, b).
In mathematical terms, we can define the limit as
follows:
Definition We say that the limit of f (x, y ) as (x, y )
approaches (a, b) is L if for every number ω > there is
a corresponding number ε > such that
!
< (x ↑ a) + (y ↑ b) < ε =↓ |f (x, y ) ↑ L| < ω
Notation
lim
(x,y )→(a,b)
f (x, y ) = L.
Recall that for y = f (x),
lim f (x) exists if and only if
x→a
lim f (x) and lim f (x) exist and lim+ f (x) = lim f (x).
x→a+
x→a→
x→a
x→a→
This was because, in R one can approach a point either
from right or left.
However, things are a bit more complicated in R .
Suppose that f (x, y ) ↔ L as (x, y ) ↔ (a, b) along a path
C and f (x, y ) ↔ L as (x, y ) ↔ (a, b) along a path C . If
L ↗= L , then
lim
f (x, y ) does not exist.
(x,y )→(a,b)
This phenomena is sometimes called as path
independence of the limit.
Definition We say that a function f (x, y ) is
continuous at (a, b) if
lim
(x,y )→(a,b)
f (x, y ) = f (a, b).
Every polynomial, exponential function is continuous.
Trigonometric functions, and logarithmic functions
are continuous in their domains. Rational functions
are continuous when their denominators are nonzero.
↑
Furthermore, a sum, difference, product or
of
continuous functions is continuous.
cx m y n
c→R
n, m → N
Example 1. Evaluate the following limits, if they exist.
↘
↘
lim
( x y ↑ xy +
lim
(e x↓y ↑ cos( x + y ) + sin( y ))
(x,y )→( ,↓ )
(x,y )→( ,↓ )
)
Example 2. For the following limits approach ( , )
along the x-axis, and along the y -axis, and along a line
passing through the origin. Does the limit exist?
x
↭
lim
(x,y )→( , ) x + y
↭
x + sin y
x +y
(x,y )→( , )
lim
↭
xy cos y
(x,y )→( , ) x + y
lim
↭
lim
(x,y )→( , )
x y
x +y
Note that the graph of z =
x y
x +y
looks like:
Example 3. Find
xy
.
(x,y )→( , ) x + y
lim
Example 4. Evaluate
xy
"
(x,y )→( , )
x +y
lim
Example 4. cont’d Evaluate
x sin y
(x,y )→( , ) x + y
lim
Example 4. cont’d Evaluate
x y
(x,y )→( , ) x + y
lim
Example 4. cont’d Evaluate
x y cos( x )
lim
(x,y )→( , ) x + y
Example 5. Evaluate
x + y + z
.
(x,y ,z)→( , , ) x + y + z
lim
Example 6. Determine the value(s) of c so that the
following function is continuous at ( , ).
!
x y
, if (x, y ) →= ( , )
x +y
f (x, y ) =
c ↑ , if (x, y ) = ( , )
Example 7. Determine whether the given function is
continuous at ( , ) or not.
f (x, y ) =
"
xy
x +xy +y
,
, if (x, y ) →= ( , )
if (x, y ) = ( , )
Example 8. Determine the set of points at which the
function is continuous.
#
cos(x ↑ y )
f (x, y ) =
, g (x, y ) = ln( + x + y ) + x + y
x
e ↑y
Example 9. Show that
x +y
= .
(x,y )→( , ) x + y
lim
Hint. Write
x +y
x x +y y
x x
y y
=
=
+
x +y
x +y
x +y
x +y
Then, split the given limit into two limits, and use the
squeeze theorem for each one.
We may want to try polar coordinates
x = r cos(ω), y = r sin(ω)
when we investigate limit problems. Notice that as
(x, y ) ↓ ( , ), we must have r ↓ .
e ↑x ↑y ↑
Example 10. Find
lim
x +y
(x,y )→( , )
Example 11. Find
lim
(x,y )→( , )
(x + y ) ln(x + y )
Remark. Consider
xy
(x,y )→( , ) x + y
lim
(↔)
By switching to the polar coordinates we obtain
r cos(ω) sin (ω)
r → cos (ω) + r sin (ω)
lim
(↔↔)
Clearly, the value of this limit is for each fixed value
of ω.
However, (↔) dne as along the parabola x = y , (↔)
becomes
y y
lim
=
y→ y + y
but along the x↑axis the limit value is .
Additional Examples
Example 12.
Evaluate
x tan(y ) ↑ y tan(x)
x +y
(x,y )→( , )
lim
Example 13.
Evaluate
x sin( y ) ↑ y sin( x)
x +y
(x,y )→( , )
lim
Example 14.
Find the value of ε so that
!
(x ↑y )
, if (x, y ) →= ( , )
x +y
f (x, y ) =
ε↑ ,
if (x, y ) = ( , )
is continuous at ( , ).
Example 15.
Evaluate
lim
(x,y )→(↑ , )
x + xy + y
x+ y
Example 16.
Evaluate
lim
(x,y )→( , )
( + xy ) /xy
Example 17.
Evaluate
(x + y )
(x,y )→( , ) x + y
lim