1. Points of condensation
Let and
We say that X is a point of condensation of M if every neigbourhood of X contains at least one point of M.

2. points of condensationM M
not a point of condensation M
a discrete point

3. Let be defined on M and let
be a point if condensation. We say that is a limit of
at A if, for every
there exists a neighbourhood such that
for all
we have

4. We use the following denotations:

5. Some basic properties of limits of functions in one variable also extend to limits of functions in more variables such as
Provided that the right-hand-side limits exist

6. Also a parallel of the squeezing lemma can be formulated:
If and and there exists a neighbourhood of A such that, for ,we have, for a function then

7. Similarly, it can be proved that, if a function has a limit at a point A, then the limit is unique.
However, finding the limit of a function in n variables or even proving that a function in n variables has a limit is a much more difficult task than in the event of a function in one variable. !

8. We will now develop some means that can help us in this difficult task and apply them to functions in two variables.

9. Let be an infinite sequence of points in .
We say that converges to a point A if, for every there is an index N such that, for
we have Formally, we write or

10. The following two assertions are equivalent:
For every sequence of points in D( f ) such that we have

11. Example
Prove that the function has a limit at [0,0]

12. Example
This limit does not exist:

13. Let be a function in two variables defined on the set
and let the limit
be defined for every
. Thus we can define
If then the
a function
number A is called the double limit of at

15. If 1
where is defined on a rectangle 2
for any there exists a limit
Then the double limit exists and

16. Example
for y = 0, no limit for x  0 exists
Counterexample showing that if condition 1 is satisfied in the above theorem condition 2 need not be.

17. Example
Reverse assertion may be false
No limit exists
different results for different approaching angles

18. We say that a function is continuous at a point
We say that a function is continuous at a point , which is a point of condensation of D( f ) , if
it has a limit at and this limit equals to the
function value at

19. If and are continuous at
then so are if

20. If and
are continuous at
, then so is the composite function

21. Note that , if a function is continuous at
its limit at is calculated
simply by substituting into