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Mathematical Expression Editor

We investigate what continuity means for functions of several variables.

Now that we have defined limits, we can define continuity.

Let and be an interior point of the domain of . We say is continuous at ,
if

exists.

exists.

is continuous on an open set if is continuous at all .

The limit laws can be used to write corresponding continuity laws.

Limit Laws Let and be continuous functions of several variables, and be a real
number.

where

Constant Law

is continuous.

Identity Law

is continuous.

Sum/Difference Law

is continuous.

Scalar Multiple Law

is continuous.

Product Law

is continuous.

Quotient Law

is continuous where .

True or false: If and are continuous functions on an open disk , then is continuous
on .

TrueFalse

True or false: If and are continuous functions on an open disk , then is continuous
on .

TrueFalse

The function may or may not be continuous, it depends on whether . If , then not
continuous at that point.

Composition Limit Law Let be a continuous function on an interval . Let be a
function whose range is contained in (or equal to) , Then

Composition of Composite Functions Let be continuous on an open disk , where the
range of on is , and let be a single variable function that is continuous on . Then is
continuous on .

Show that the function is continuous for all points in .

Let Since is not
actually used in the function, and polynomials are continuousare not
continuous, we conclude is continuous everywhere. A similar statement can be made about
Setting we obtain a continuous function from . Since sine is continuousis not
continuous for all real values, the composition of sine with is continuous. Hence, is continuous
everywhere. We finish by presenting you with a plot of :

Let Is continuous at ? Is continuous everywhere?

To determine if is continuous at ,
we need to compare Applying the definition of , we see that: We now consider the
limit Substituting for and in returns the indeterminate form , so we need to do
more work to evaluate this limit.

Consider two related limits:

The first limit does not contain , and since is continuous,

The second limit does not contain . But we know

Finally, we know that we can combine these two limits so that

We have found that , so is continuous at .

A similar analysis shows that is continuous at all points in . As long as , we can
evaluate the limit directly; when , a similar analysis shows that the limit is . Thus we
can say that is continuous everywhere. We finish by presenting you with a plot of :