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->x₀f(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) = x^1/n are continuous in their proper domain since lim x -> x₀ (x^1/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) = e^x 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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HOW TO EXAMINE THE CONTINUITY OF THE FUNCTION

Points to Remember While Checking Continuity of a Functions

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