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.

points of condensation M M not a point of condensation M a discrete point

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

We use the following denotations:

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

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

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. !

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

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

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

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

Example This limit does not exist:

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

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

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.

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

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

If and are continuous at then so are if

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

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