In
Continuity , we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. In particular, three conditions are necessary for
to be continuous at point
exists.
exists.
These three conditions are necessary for continuity of a function of two variables as well.
Definition
A function
is continuous at a point
in its domain if the following conditions are satisfied:
exists.
exists.
Demonstrating continuity for a function of two variables
Show that the function
is continuous at point
There are three conditions to be satisfied, per the definition of continuity. In this example,
and
exists. This is true because the domain of the function
consists of those ordered pairs for which the denominator is nonzero (i.e.,
Point
satisfies this condition. Furthermore,
exists. This is also true:
This is true because we have just shown that both sides of this equation equal three.
Continuity of a function of any number of variables can also be defined in terms of delta and epsilon. A function of two variables is continuous at a point
in its domain if for every
there exists a
such that, whenever
it is true,
This definition can be combined with the formal definition (that is, the
epsilon–delta definition ) of continuity of a function of one variable to prove the following theorems:
The sum of continuous functions is continuous
If
is continuous at
and
is continuous at
then
is continuous at
The product of continuous functions is continuous
If
is continuous at
and
is continuous at
then
is continuous at
The composition of continuous functions is continuous
Let
be a function of two variables from a domain
to a range
Suppose
is continuous at some point
and define
Let
be a function that maps
to
such that
is in the domain of
Last, assume
is continuous at
Then
is continuous at
as shown in the following figure.
The composition of two continuous functions is continuous.
Let’s now use the previous theorems to show continuity of functions in the following examples.
More examples of continuity of a function of two variables
Show that the functions
and
are continuous everywhere.
The polynomials
and
are continuous at every real number, and therefore by the product of continuous functions theorem,
is continuous at every point
in the
Since
is continuous at every point
in the
and
is continuous at every real number
the continuity of the composition of functions tells us that
is continuous at every point
in the
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The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
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