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->xf(x) exists.

(iii) f(x0)  =  lim x -> x0 f(x)

Points to Remember While Checking Continuity of a Functions

(1)  Constant function is continuous at each point of R (R stands for real numbers)

(2)  Power functions with positive integer exponents are continuous at every point of R

(3)  Polynomial functions, p(x) are continuous at every point of R.

(4)  Quotients of polynomials namely rational functions of the form

R(x) = p(x) / q(x), are continuous at every point where q(x) ≠ 0, and

(5)  The circular functions sin x and cos x are continuous at every point of their domain  R  = (- ∞, ∞) since lim x-> x0 sin x = sin x0, lim x-> x0 cos x  =  cos x0

(6)  The nth root functions, f (x) = x1/n are continuous in their proper domain since lim x -> x0 (x1/n)  =  x0 1/n

(7)  The reciprocal function f (x) = 1/x is not defined at 0 and hence it is not continuous at 0. It is continuous at each point of R − {0} .

(8)  The exponential function f(x) = ex is continuous on R.

(9) The logarithmic function f(x) = log x (x > 0) in continuous in (0, ∞)

(10)  The modulus function

f(x) = |x|

-x if x < 0

0 if x = 0

x if x > 0 

is continuous at all points of the real line R .

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"

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