Limits and Continuity of Functions of Two or More
Variables
Introduction
Recall that for a function of one variable, the mathematical statement
means that for x close enough to c, the difference between f(x) and L is "small".
Very similar definitions exist for functions of two or more variables; however,
as you can imagine, if we have a function of two or more independent
variables, some complications can arise in the computation and interpretation
of limits. Once we have a notion of limits of functions of two variables we can
discuss concepts such as
continuity
and
derivatives
.
Limits
The following definition and results can be easily generalized to functions of
more than two variables. Let f be a function of two variables that is defined in
some circular region around (x_0,y_0). The limit of f as x approaches
(x_0,y_0) equals L if and only if for every epsilon>0 there exists a delta>0 such
that f satisfies
whenever the distance between (x,y) and (x_0,y_0) satisfies
We will of course use the natural notation
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 Spring '08
 Staff
 Calculus, Continuity, Derivative, Limits, Continuous function

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