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) limx →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
Br ((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.
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