. . Recall of Limit and Continuity of Functions of One Variable Next we shall look at continuity of a function of two variables. Before that, it will be beneficial for us to recall the continuity of a function of single variable.
We have seen the following definition of continuity in XI Std. A function f a b :( , ) → is said to be continuous at a point x a b ∈ ( , ) if the following hold: ( ) f is defined at x . ( ) lim x f x L exists ( ) L The key idea in the continuity lies in understanding the second condition given above. We write lim x f x L whenever the value f x ( ) gets closer and closer to L as x gets closer and closer to x .
To make it clear and precise, let us rewrite the second condition in terms of neighbourhoods. This will help us when we talk about continuity of functions of two variables. Definition . (Limit of a Function) Suppose that f a b :( , ) → and x a b ∈ ( , ) .
We say that f has a limit L at x if for every neighbourhood ( ), L L > ε ε ε of L , there exists a neighbourhood ( ( , ), a b ⊂ > δ δ δ of x such that L L ε ε whenever x ) \{ } δ δ The above condition in terms of neighbourhoods can also be equivalently stated using modulus (or absolute value) notation as follows: and and such that | | L whenever | | δ . This means whenever x ¹ and is within δ distance from x , then f x ( ) is within distance from L . Following figures explain the interplay between ε and δ . Fig.
. Fig. . We also know, from XI