FIND THE INTERVALS ON WHICH EACH FUNCTION IS CONTINUOUS

About "Find the Intervals on Which Each Function is Continuous"

Find the Intervals on Which Each Function is Continuous :

Here we are going to how to examine the continuity of the function when the interval is not given.

Three requirements have to be satisfied for the continuity of a function y = f(x) at x = x0 :

(i) f(x) must be defined in a neighbourhood of x0 (i.e., f(x0) exists);

(ii) lim x->xf(x) exists.

(iii) f(x0)  =  lim x -> x0 f(x)

To know the points to be remembered in order to decide whether the function is continuous at particular point or not, you may look into the page " How to Check Continuity of a Function If Interval is not Given "

Question 1 :

Examine the continuity of the following 

sin x / x2

Solution :

Let f(x)  =  sin x / x2

(i)  From the given function, we know that both "x" and "sin x " are defined for all real numbers. Here "x" is in the denominator, so it should not be 0.

(ii)  lim x-> x0 f(x)  =  lim x-> x0 sin x / x2

By applying the limit, we get

 =  sin x0 / (x0)2 -------(1)

(iii) f(x0)  =   sin x0 / (x0)2 -------(2)

From (1) and (2)

lim x-> x0 f(x)  =  f(x0)

Hence the given function is continuous for all ∈ R - {0}.

Question 2 :

Examine the continuity of the following 

(x2 - 16) / (x + 4)

Solution :

Let f(x)  =  (x- 16) / (x + 4)

If we apply x = -4, then the denominator will become zero. So the entire value will become infinity.

Hence it is continuous for all ∈ R - {-4}.

Question 3 :

Examine the continuity of the following 

|x + 2| + |x - 1|

Solution :

Since we have a absolute value function, we have to split into piece wise function and check its continuity.

x + 2  =  0

x  =  -2

x - 1  =  0

x  =  1

f(x)  =  -x - 2 - x + 1  =  -2x - 1     If x < -2

f(x)  =  x + 2 - x + 1  =  3      If -2 ≤ x < 1

f(x)  =  x + 2 + x - 1  =  2x + 1   If x ≥ 1

From the above piece wise unction, we have to check if it is continuous at x = -2 and x = 1

lim x->-2- f(x) =  -2(-2) - 1

  =  4 - 1  

  =  3 ---(1)

lim x->-2+ f(x) =  3 ---(2)

Since left hand limit and right hand limit are equal for -2, it is continuous at x = -2.

lim x->1- f(x) =  3

lim x->1+ f(x) =  2x + 1

  =  2(1) + 1

  =  3

Since left hand limit and right hand limit are equal for 1, it is continuous at x = 1.

Hence the function is continuous for x ∈ R

After having gone through the stuff given above, we hope that the students would have understood, "Find the Intervals on Which Each Function is Continuous"

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