Functions of two or more Variables

Functions of two or more Variables:

Suppose D is a set of n- tuples of real numbers
A real-valued function f on D is a rule that assigns a real number to each element in D. The set D is the
Function s domain. The set of w-values taken on by f is the function s range. The symbol w is the dependent variable of f and f is said to be a function of the n independent variables
We also call the x s the function s input variables and
call w the function s output variable.

Example:
at the The value of the function
point (3,0,4) is:


Example:
is a function of two variables.
is a function of three variables.
Limits and Continuity:
is the number L if for as The limit of
such that for all points there exists a any
in the domain of f, either:
or (1)
. (2)

Remark:
For functions of a single variable:
(any number
k).
Properties of Limits of Functions of two Variables:

, then: and If
(1)

(2)

(any number k) (3)

(4)

Example:
(1)

(2)


Continuity:
is said to be continuous at the point A function
If:
. (1) f is defined at

exists. (2)

. (3)
A function is said to be continuous if it is continuous at every point of its domain .