2.2
Limits and continuity
In order to introduce the concepts of limit and continuity for functions of more than one variable
we need first to generalise the concept of neighbourhood of a point from R to R2 and R3 .
Let us start by recalling the corresponding definition for functions of one variable.
2.2.1
Neighbourhood of a point in one, two and three dimensions
Definition: If x0 is a point in the real line and ε > 0, then the set of points x in the real line
whose distances from x0 are less than ε is called an open interval about x0 or a neighbourhood of x0 . If we denote this neighbourhood by Nε (x0 ), then
Nε (x0 ) = {x : |x − x0 | < ε}
(2.9)
Definition: If (x0 , y0 ) is a point in the 2-dimensional xy-plane and ε > 0, then the set of
points (x, y) in the plane whose distances from (x0 , y0 ) are less than ε is called an open disk
of radius ε about (x0 , y0 ) or, more simply, a neighbourhood of (x0 , y0 ). If we denote this
neighbourhood by Nε ((x0 , y0 )), then
n
o
p
Nε ((x0 , y0 )) = (x, y) : (x − x0 )2 + (y − y0 )2 < ε .
(2.10)
Note: Since the equation
(x − x0 )2 + (y − y0 )2 = ε2 ,
(2.11)
defines a circle of radius ε centered at the point (x0 , y0 ), the neighbourhood of (x0 , y0 ) are all
the points contained inside that circle (that is, a disk) but not the points on the circle.
Figure 3: Neighbourhood of a point in one, two and three dimensions. The regions D are arbitrary
subspaces of R2 or R3 where the point and its neighbourhood live.
Definition: If (x0 , y0 , z0 ) is a point in the 3-dimensional xyz-plane and ε > 0, then the set of
points (x, y, z) in the plane whose distances from (x0 , y0 , z0 ) are less than ε is called an open
ball of radius ε about (x0 , y0 , z0 ) or, more simply, a neighbourhood of (x0 , y0 , z0 ). If we
denote this neighbourhood by Nε ((x0 , y0 , z0 )), then
n
o
p
Nε ((x0 , y0 , z0 )) = (x, y, z) : (x − x0 )2 + (y − y0 )2 + (z − z0 )2 < ε .
(2.12)
8
Note: Since the equation
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = ε2 ,
(2.13)
defines a sphere of radius ε centered at the point (x0 , y0 , z0 ), the neighbourhood of (x0 , y0 , z0 )
are all the points contained inside that sphere (that is, a ball) but not the points lying on the
sphere.
It is convenient here to introduce some notions related to sets in the 2-dimensional space:
Definition: Let S be a set of points in the plane:
• We say that a point (x0 , y0 ) is a boundary point of S if every neighbourhood of (x0 , y0 )
contains at least one point in S and at least one point not in S.
• We say that S is closed (or is a closed set) if every boundary point of S belongs to S.
• We say that S is open (or is an open set) if no boundary point of S belongs to S.
• The interior of S is the set of all points in S that are not boundary points of S. The
exterior of S is the set of all points that are not in S and not boundary points of S.
Figure 3: Boundary points, interior and exterior points corresponding to the set S defined by x2 +y 2 ≤ 1,
that is the closed disk of radius 1.
2.2.2
Limits of functions of two variables
The concept of limit of a function of several variables is similar to that for functions of one
variable. For clarity we will present here only the case of functions of 2 variables; the general
case is a natural generalization of that one.
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Definition: We say that a function f (x, y) approaches the limit L as the point (x, y) approaches the point (x0 , y0 ) and we write
lim
(x,y)→(x0 ,y0 )
f (x, y) = L,
(2.14)
if all points of a neighbourhood of (x0 , y0 ), except possibly the point (x0 , y0 ) itself, belong to
the domain D(f ) of f , and if f (x, y) approaches L as (x, y) approaches (x0 , y0 ). In other words,
the closer the point (x, y) is to (x0 , y0 ), the closer is the function f (x, y) to the limiting value L.
A more formal definition of limit: We say that
lim
(x,y)→(x0 ,y0 )
f (x, y) = L,
(2.15)
• every neighbourhood of (x0 , y0 ) contains points (x, y) in D(f ), and
• if and only if for every number ε > 0 there exists another number δ(ε) > 0 such that
p
(2.16)
if 0 < (x − x0 )2 + (y − y0 )2 < δ(ε) then |f (x, y) − L| < ε,
whenever (x, y) ∈ D(f ).
Laws of limits: All usual laws of limits extend to functions of several variables in the obvious
way. For example if
lim
(x,y)→(x0 ,y0 )
f (x, y) = L and
lim
(x,y)→(x0 ,y0 )
g(x, y) = M,
(2.17)
then
lim
(f (x, y) ± g(x, y)) = L ± M
(2.18)
lim
f (x, y)g(x, y) = LM,
(2.19)
(x,y)→(x0 ,y0 )
(x,y)→(x0 ,y0 )
f (x, y)
L
=
.
M
(x,y)→(x0 ,y0 ) g(x, y)
lim
(2.20)
Existence of the limit:
• If a limit exists it is unique.
• For a function of one variable f (x) the existence of the limit limx→x0 f (x) implies that the
function f (x) approaches the same finite number as x approaches x0 from either the right
or the left.
• For a function of 2 variables lim(x,y)→(x0 ,y0 ) f (x, y) = L exists only if f (x, y) approaches the
same number L no matter how (x, y) approaches (x0 , y0 ) in the xy-plane. That means
that (x, y) can approach (x0 , y0 ) along any curve in the xy-plane which goes through
(x0 , y0 ).
• Notice that it is not necessary that f (x0 , y0 ) = L, even if f (x0 , y0 ) is defined.
Let us now exemplify these definitions with various examples:
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Example 1: Let us consider the following limits
lim
(x,y)→(1,1)
2x − y 2 = 2 − 1 = 1,
lim
(x,y)→(x0 ,y0 )
lim
(x,y)→(π,2)
x2 y = x20 y0 ,
y sin(xy) = 2 sin(2π) = 0.
(2.21)
(2.22)
(2.23)
All these examples show limits which exist, in the sense that the functions approach the
same limit no matter how (x, y) approaches the limiting point. In fact, in all these cases
the limit of the function coincides with the value of the function at the limiting point. As
we said above this is not always the case.
Example 2: Let
f2 (x, y) =
2xy
,
+ y2
(2.24)
f2 (x, y).
(2.25)
x2
and try to compute the limit
lim
(x,y)→(0,0)
It is not difficult to prove that this limit does not exist. This is due to the fact that the
function f (x, y) approaches different limiting values depending on the way the point (x, y)
approaches (0, 0). To see that, suppose that we take the limit (x, y) → (0, 0) along curves
parameterized by y = kx, where k is an arbitrary real number different from 0,
lim
(x,kx)→(0,0)
f2 (x, kx) =
2kx2
2k
=
.
2
2
2
1 + k2
(x,kx)→(0,0) x + k x
lim
(2.26)
Obviously, the value of the limit is different for each different value of k. Therefore the
limit is not unique which implies it does not exist. This is also an example of a function
which is not well defined at the limiting point (0, 0), since f (0, 0) = 0/0 = indeterminate.
Example 3: Let
f3 (x, y) =
2x2 y
.
x4 + y 2
(2.27)
lim
f3 (x, y).
(2.28)
We want to compute the limit
(x,y)→(0,0)
We can try to compute the limit along lines parameterized as before, y = kx with k 6= 0.
We obtain
2kx
2kx3
= lim 2
= 0.
(2.29)
lim
f3 (x, kx) = lim 4
2
2
x→0 x + k 2
x→0 x + k x
(x,kx)→(0,0)
So the limit along any straight line y = kx is 0. We might be then tempted to conclude
that lim(x,y)→(0,0) f (x, y) = 0, but this would be incorrect. To see that let us now take the
limit along curves of the form y = kx2 . Then
lim
(x,kx3 )→(0,0)
2kx4
2k
=
.
4
2
4
x→0 x + k x
1 + k2
f3 (x, kx2 ) = lim
(2.30)
So, the limit along curves y = kx2 is not 0 but gives a different value for each different
choice of k. Therefore the limit (2.28) does not exist.
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Example 4: Let
x2 y
.
x2 + y 2
(2.31)
f4 (x, y) = 0.
(2.32)
f4 (x, y) =
and prove that the limit
lim
(x,y)→(0,0)
We can try to prove that by choosing some particular curves along which to take the limit,
as in the previous examples. However, to be really sure that the limit is always the same,
no matter which curve we take, we can try to prove (2.32) from the rigorous definition of
the limit given in (2.16). Since x2 ≤ x2 + y 2 and y 2 ≤ x2 + y 2 , we have that
¯
¯
p
¯ x2 y ¯
¯ ≤ |y| ≤ x2 + y 2 .
¯
|f4 (x, y) − 0| = ¯ 2
(2.33)
¯
2
x +y
So, formally if we consider a number ε > 0 and we take δ(ε) = ε it is guaranteed that if
p
(2.34)
0 < x2 + y 2 < ε,
then
|f4 (x, y) − 0| < ε,
(2.35)
therefore the limit according to definition (2.16) exists and is (2.32).
Figure 4: Graphic representation of the function f4 (x, y) = x2 y/(x2 + y 2 ).
2.2.3
Continuity of functions of two variables
Definition: The function f (x, y) is continuous at the point (x0 , y0 ) if
lim
(x,y)→(x0 ,y0 )
f (x, y) = f (x0 , y0 ).
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(2.36)
Note: Notice that it is not always true that the limit of a function at a certain point
coincides with the value of the function at that point. Therefore the condition (2.36) is a
non-trivial constraint. For example, we can define the following function
½
0 if (x, y) 6= (0, 0)
f (x, y) =
(2.37)
1 if (x, y) = (0, 0)
It is clear that
lim
(x,y)→(0,0)
f (x, y) = 0,
(2.38)
since when performing the limit we approach the point (0, 0) “infinitely” but we never exactly
reach it. Therefore the value of the limit is in this case the value of the function everywhere but
at (0, 0), which is by definition 0. However f (0, 0) = 1. Therefore, this function is discontinuous
at (0, 0). We can easily make this function continuous everywhere by simply changing its value
at the origin to f (0, 0) = 0.
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