Std, that a function f defined in the neighbourhood of x except possibly at x has a limit at x if the following hold : ( ) lim L (right hand limit) exists ( ) lim L (left hand limit) exists ( ) L = L . Let L (say). Then the function f is continuous at x= x if L = L = L .Note that in the limit and continuity of a single variable functions, neighbourhoods play an important role. In this case a neighbourhood of a point x ∈ looks like ( r x , where r > .
In order to develop limit and continuity of functions of two variables, we need to define neighbourhood of a point ( , ) u v ∈ . So, for ( , ) u v ∈ and r > , a r -neighbourhood of the point ( , ) u v is the set B u v x y v r (( , )) {( , ) | ( } . So a r -neighbourhood of a point ( , ) u v is an open disc with centre ( , ) u v and radius r > . If the centre is removed from the neighbourhood, then it is called a deleted neighbourhood .