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*) →
ℝ is said to be continuous at a point
*x*_{0} ∈(*a*,
*b*) if the following hold:

(1) *f* is defined at *x _{0}* .

(2)
lim_{ x }_{→x0 } *f (x)*
= L exists

(3) L* = f(x*_{0})

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 *x*_{0}.

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 *x*_{0} ∈(*a*, *b*) . We say that *f* has a limit *L* at *x* = *x*_{0} if for
every neighbourhood (*L* − *ε* , *L* + *ε* ), *ε* > 0 of *L*
, there xists a neighbourhood ( *x*_{0}
− *δ* , *x*_{0} + *δ* ) ⊂ ( *a* , *b*), *δ* > 0 of *x*_{0}
such that

* f*( *x*) ∈ ( *L* − *ε* , *L* + *ε* ) whenever *x* ∈ ( *x*_{0} −*δ* , *x*_{0} + *δ* ) \ {*x*_{0} } .

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 *− x_{0} | < δ .

This
means whenever *x* ≠
*x*_{0} and is within *δ* distance from *x*_{0} , 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 x_{0} except possibly at *x*_{0} has a limit at *x*_{0}
if the following hold :

(1) lim_{
x →x0+} *f (x)* = L_{1} (right
hand limit) exists

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

(3) L_{1}
= L_{2}.

Let *f (x _{0})* = L (say). Then the
function

*B _{r} *((

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

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