HOW TO EXAMINE THE CONTINUITY OF THE FUNCTION

About "How to Check Continuity of a Function If Interval is not Given"

How to Check Continuity of a Function If Interval is not Given :

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)

Points to Remember While Checking Continuity of a Functions

Question 1 :

Prove that f(x) = 2x² + 3x - 5 is continuous at all points in R.

Solution :

Since the given function f(x) is a polynomial, it is continuous at all points in R.

Question 2 :

Examine the continuity of the following 

x + sin x

Solution :

Let f(x)  =  x + sin x

(i)  From the given function, we come to know that both "x" and "sin x" are defined for all real numbers.

(ii)  lim _{x-> x0} f(x)  =  lim _{x-> x0} x + sin x

By applying the limit, we get

 =  x₀ + sin x₀ -------(1)

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

From (1) and (2)

lim _{x-> x0} f(x)  =  f(x₀)

Hence the given function is continuous for all real values.

After having gone through the stuff given above, we hope that the students would have understood, "How to Check Continuity of a Function If Interval is not Given"

Apart from the stuff given in "How to Check Continuity of a Function If Interval is not Given",  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.