Circular and Rectangular Neighbourhoods 

Examples 
Ex 1: Let f(x, y) = ; (x, y) ≠ (0, 0). Find
i) f(x, y)
ii) f(x, y). 
Sol:
i) = =
ii) = = 
Ex 2: Determine whether the simultaneous limit , where f(x, y) = ; (x, y) ≠ (0, 0) exists or not? 
Sol: We know that if the simultaneous limit of a function exists, then the two repeated limits also exist and all the three limits have the same value.
= 3 = 3
= =
Thus both the repeated limits exist but are not equal. Hence, the simultaneous limit of f at (0, 0) does not exist. 
Introduction
Let us consider the concepts of limits, continuity and differentiability of functions of two (or more) real variables. Functions with two (or more) independent variables are common in science. Their derivatives are interesting and their integrals lead to a variety of applications.
The Cartesian product A × B of sets A and B is the collection of all ordered pairs (a, b) with a ∈ A and b ∈ B. If A ⊆ R and B ⊆ R, A × B is a subset of R × R, i.e., R^{2} whose elements are considered to be points in the 2D plane. If (a, b) ∈ R^{2}, it is associated with a unique point in the 2D plane with Cartesian coordinates (a, b).
If we denote A × B = E, so that E ⊂ R^{2}, then the function f : E → R is termed a real valued function of two variables. For (x, y) ∈ E, it's image under f is denoted by f(x, y) or 'z' (a real number). Henceforth, a function of two variables means a real valued function of two real variables.
Basic Concepts
Circular neighbourhood:
Let (a, b) ∈ R^{2}. For a positive number δ, the set {(x, y) ∈ R^{2} : < δ} is called the δneighbourhood of (a, b) or as the circular neighbourhood of (a, b) with radius δ. Refer adjacent figure.
Rectangular neighbourhood:
Let (a, b) ∈ R^{2}. For two positive numbers δ_{1} and δ_{2}, the set {(x, y) ∈ R^{2} : x – a < δ_{1} and y – b < δ_{2}} is called the rectangular neighbourhood of (a, b). Refer adjacent figure.
Note that every rectangular neighbourhood of (a, b) contains a circular neighbourhood of (a, b). The vice–versa is also true.
Simultaneous limit:
A function f : E → R is said to tend to a limit 'l' as (x, y) approaches (a, b).
Symbolically .
Note that the difference between 'l' and the values taken by the function has to be made arbitrarily small for all points close to (a, b). In other words, ⇔ for a given ∈ > 0, there exists a δ > 0 such that, for all (x, y) ∈ E with 0 < < δ, we have f(x, y) – l < ∈. Then 'l' is called the simultaneous limit of f as (x, y) nears (a, b).
Repeated limit:
Let f : E → R be a function of two variables, where E ⊂ R^{2}.
For any x ∈ R, let E_{x} = {y ∈ R : (x, y) ∈ E} so that E_{x} ⊂ R.
For a given 'x', f is considered as a realvalued function on E_{x}.
If for each 'x', the following two limits exist:
(i) and
(ii)
then 'α' is called a repeated limit.
It is denoted by
Similarly, we can also define the second repeated limit .
Note:
i) If the simultaneous limit of a function exists, then the two repeated limits also exist. And all the three limits have the same value.
ii) But if the repeated limits exist and are equal, the simultaneous limit may or may not exist.
Continuity:
The function f : E → R is said to be continuous at (a, b) ∈ E, if exists and is equal to f(a, b).
That is, f : E → R is continuous at (a, b) ∈ E if, for every ∈ > 0, there exists a δ > 0 such that for all (x, y) ∈ E, with 0 < < δ, we have f(x, y) – f(a, b) < ∈ or = f(a, b).