. Limit and Continuity of Functions of Two Variables Definition . (Limit of a Function) Suppose that A x y a b c d F A : . We say that F has a limit L at u v if the following hold : For every neighboourhood ( ), L L > , of L , there exists a δ -neighbourhood B u v A δ (( , )) ⊂ of ( , ) u v such that ( , ) (( , )) \{( , )}, x y B u v u v L L > ⇒ d d We denote this by lim x y u v F x y L if such a limit exists.
O – – – – – – – – – – – – – – and - - Differentials and Partial Derivatives Fig. . Limit of a function When compared to the case of a function of single variable, for a function of two variables, there is a subtle depth in the limiting process. Here the values of F x y ( , ) should approach the same value L , as ( , ) x y approaches ( , ) u v along every possible path to ( , ) u v (including paths that are not straight lines).
Fig. . explains the limiting process. All the rules for limits (limit theorems) for functions of one variable also hold true for functions of several variables .
Now, following the idea of continuity for functions of one variable, we define continuity of a function of two variables. Definition . (Continuity) Suppose that A x y a b c d F A