Limit and Continuity of Functions of Two Variables: Definition 8.6 (Limit of a Function), Definition 8.7 (Continuity)

**Limit and
Continuity of Functions of Two Variables**

Suppose that A {(* x *,
y) | *a* <* x < *b , c* < y < d*
} ⊂ ℝ^{2} , F : A → **R**. We say
that F has a limit L at (*u,v*) if the
following hold :

For every neighboourhood ( L − ε , L + ε ), ε > 0 , of L , there
exists a δ –neighbourhood B_{δ} ((u,*
v *)) ⊂ A of (*u,v*) such that
(* x *, y) ∈ B_{δ} ((u,* v *))
\ {(u,* v *)}, δ > 0 ⇒ *f (x)* ∈(L − ε , L + ε ) .

We denote this by lim_{ (
x , y ) →(u,v)} F ( x,* y *) = L
if such a limit exists.

When
compared to the case of a function of single variable, for a function of two
variables, there is a subtle depth in the limiting process. Here the values of *F* ( *x*,
*y*) should approach the same value *L* , as ( *x* , *y*) approaches (*u* , *v*)
along **every
possible path to** (*u* , *v*) (including paths that are not
straight lines). Fig.8.9 explains the limiting process.

**All the rules for
limits (limit theorems) for functions of one variable also hold true for
functions of several variables**.

Now,
following the idea of continuity for functions of one variable, we define
continuity of a function of two variables.

Suppose that *A* = {( *x*, *y* ) | *a* < *x* < *b*, *c* < *y* < *d* } ⊂ ℝ^{2} , *F* : *A* → ℝ. We say that *F* is
continuous at at (*u* , *v*) if the following hold :

(1)* F* is defined at (*u* , *v*)

(2) lim_{( x , y ) →(u,v)} F ( x,* y *) = L
exists

(3) L = F (*u,v*).

**Remark**

(1) In
Fig. 8.10 taking *L* =
*F* ( *x*_{0} , *y*_{0}
) will illustrate continuity at ( *x*_{0}
, *y*_{0} ) .

(2) Continuity
for *f* (*x*_{1} , *x*_{2}
, , *x _{n}* ) is also defined
similarly as defined above.

Let us
consider few examples as illustrations to understand continuity of functions of
two variables.

** **

**Example 8.8**

Let* f *(*x
, y*) = (or) 3x−5*y*+8 / x^{2}+y^{2}+1
for all (* x *, y) ∈ ℝ^{2}. Show that* f *is continuous on ℝ^{2}.

**Solution**

Let (a ,
b) ∈ ℝ^{2} be an arbitrary point. We shall
investigate continuity of* f *at (a ,
b).

That is,
we shall check if all the three conditions for continuity hold for* f *at (a , b) .

To check
first condition, note that* f (a, b)* = is defined.

Now we
note that lim_{x,y}_{→( a , b)}* f *(x,* y *) = L =* f (a, b)* . Hence* f *satisfies all the three conditions for continuity of* f *at (a , b) . Since (a , b) is an
arbitrary point in ℝ^{2} , we conclude that* f *is continuous at every point of ℝ^{2}.

** **

**Example 8.9**

Consider
*f* ( *x*, *y*) = if (*
x *, y) ≠ (0, 0) and *f *(0, 0)* *=* *0*
*. Show that* f *is not continuous
at (0, 0) and continuous at all other points of ℝ^{2 }.

**Solution**

Note
that *f* is defined for every ( *x* , *y*)
∈ **R**^{2} . First let us check the continuity at ( *a* , *b*)
≠
(0, 0) .

Let us
say, just for instance, (*a* , *b*) = (2, 5) . Then *f* (2, 5) = 10/29 . Then, as in the above
example, we calculate

it follows that* f *is continuous at (2, 5) .

Exactly by
similar arguments we can show that* f *is
continuous at every point *( a , b)* ≠
(0, 0) . Now let us check the continuity at (0, 0) . Note that* f *(0, 0) = 0 by definition. Next we want
to find if

exists
or not.

First
let us check the limit along the straight lines* y *=* mx *, passing through
(0, 0).

So for
different values of* m *, we get
different values* m / *1+ m^{2 }and
hence we conclude that does not exist. Hence* f *cannot be continuous at (0, 0) .

**Example 8.10**

Consider* g *( *x
, y*) = if (* x *, y) ≠
(0, 0) and g(0, 0) = 0. Show that* g *is
continuous on ℝ^{2}.

**Solution**

Observe
that the function *g* is defined for
all ( *x* , *y*) ∈
ℝ^{2} . It is easy to check, as in the
above examples, that *g* is continuous
at all point ( *x* , *y*) ≠ (0, 0) . Next we shall check the
continuity of *g* at (0, 0) . For that
we see if *g* has a limit *L* at (0, 0) and if *L* = *g*(0, 0) =
0 . So we consider

Note
that in the final step above we have used 2|*xy*|
≤* x*^{2} + y^{2} (which
follows by considering 0 ≤ (*x* − *y*)^{2} )
for all *x* , *y* ∈ ℝ. Note that ( *x* , *y*) →
(0, 0) implies |*x|* →
0 . Then from (9) it follows that * *= 0 = g(0, 0) ; which proves that* g *is continuous at (0, 0) . So* g *is continuous at every point of ℝ^{2}.

Tags : Mathematics , 12th Maths : UNIT 8 : Differentials and Partial Derivatives

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12th Maths : UNIT 8 : Differentials and Partial Derivatives : Limit and Continuity of Functions of Two Variables | Mathematics

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