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->x₀  f(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 x ∈ R - {0}.

Question 2 :

Examine the continuity of the following 

(x² - 16) / (x + 4)

Solution :

Let f(x)  =  (x² - 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 x ∈ 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.

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

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

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"

Apart from the stuff given in "Find the Intervals on Which Each Function is Continuous",  if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.