HOW TO EXAMINE THE CONTINUITY OF THE FUNCTION

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.

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