Recall of Limit and Continuity of Functions of One Variable

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) → ℝ is said to be continuous at a point x₀ ∈(a, b) if the following hold:

(1) f is defined at x₀ .

(2) lim _{x →x0}  f (x) = L exists

(3) L = f(x₀)

The key idea in the continuity lies in understanding the second condition given above. We write lim _{x →x0} f (x) = L whenever the value f ( x) gets closer and closer to L as x gets closer and closer to x₀.

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 8.5 (Limit of a Function)

Suppose that f : (a, b) → ℝ and x₀ ∈(a, b) . We say that f has a limit L at x = x₀ if for every neighbourhood (L − ε , L + ε ), ε > 0 of L , there xists a neighbourhood ( x₀ − δ , x₀ + δ ) ⊂ ( a , b), δ > 0 of x₀ such that

 f( x) ∈ ( L − ε , L + ε ) whenever x ∈ ( x₀ −δ , x₀ + δ ) \ {x₀ } .

The above condition in terms of neighbourhoods can also be equivalently stated using modulus (or absolute value) notation as follows:

∨ ε > 0, ∃ δ  > 0 such that | f (x) -  L | < ε whenever 0 < | x − x₀ | < δ .

This means whenever x ≠ x₀ and is within δ distance from x₀ , then f ( x) is within ε distance from L . Following figures explain the interplay between ε and δ .

We also know, from XI Std, that a function f defined in the neighbourhood of x₀ except possibly at x₀ has a limit at x₀ if the following hold :

(1) lim _{x →x0+} f (x) = L₁ (right hand limit) exists

(2) lim_{x →x0−} f (x) = L₂ (left hand limit) exists

(3) L₁ = L₂.

Let f (x₀) = L (say). Then the function f is continuous at x= x₀ if L = L₁ = L₂ .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 x₀ ∈ ℝ looks like ( x₀ − r, x₀ + r) , where r > 0 . In order to develop limit and continuity of functions of two variables, we need to define neighbourhood of a point (u , v) ∈ ℝ². So, for (u , v) ∈ ℝ² and r > 0 , a r -neighbourhood of the point (u , v) is the set

B_r ((u, v )) = {( x, y ) ∈ ℝ² | ( x − u ) ² + ( y − v ) ² < r² }.

So a r -neighbourhood of a point (u , v) is an open disc with centre (u , v) and radius r > 0 . If the centre is removed from the neighbourhood, then it is called a deleted neighbourhood.