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# Recall of Limit and Continuity of Functions of One Variable

Next we shall look at continuity of a function of two variables.

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) ŌåÆ ŌäØ is said to be continuous at a point x0 Ōłł(a, b) if the following hold:

(1) f is defined at x0 .

(2) lim x ŌåÆx0  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 ŌåÆx0 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) ŌåÆ ŌäØ and x0 Ōłł(a, b) . We say that f has a limit L at x = x0 if for every neighbourhood (L ŌłÆ ╬Ą , L + ╬Ą ), ╬Ą > 0 of L , there xists a neighbourhood ( x0 ŌłÆ ╬┤ , x0 + ╬┤ ) ŌŖé ( a , b), ╬┤ > 0 of x0 such that

f( x) Ōłł ( L ŌłÆ ╬Ą , L + ╬Ą ) whenever x Ōłł ( x0 ŌłÆ╬┤ , x0 + ╬┤ ) \ {x0 } .

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

Ōł© ╬Ą > 0, Ōłā ╬┤  > 0 such that | f (x) -  L | < ╬Ą whenever 0 < | x ŌłÆ x0 | < ╬┤ .

This means whenever x ŌēĀ x0 and is within ╬┤ distance from x0 , then f ( x) is within ╬Ą distance from L . Following figures explain the interplay between ╬Ą and ╬┤ . 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 ŌåÆx0+ f (x) = L1 (right hand limit) exists

(2) limx ŌåÆx0ŌłÆ 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 Ōłł ŌäØ looks like ( x0 ŌłÆ 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) Ōłł ŌäØ2. So, for (u , v) Ōłł ŌäØ2 and r > 0 , a r -neighbourhood of the point (u , v) is the set

Br ((u, v )) = {( x, y ) Ōłł ŌäØ2 | ( x ŌłÆ u ) 2 + ( y ŌłÆ 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.

Tags : Functions of Several Variables | Mathematics , 12th Maths : UNIT 8 : Differentials and Partial Derivatives
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12th Maths : UNIT 8 : Differentials and Partial Derivatives : Recall of Limit and Continuity of Functions of One Variable | Functions of Several Variables | Mathematics