Find the points of discontinuity of f , where f(x) = {(sinx/x, if x < 0) (x+1, if x ≥ 0)

Solution:

The given function is f(x) = {(sin x / x, if x < 0) (x + 1, if x ≥ 0)

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

If c < 0, then f(c) = sin c f(c)

= sin⁡ c

lim_x→c f(x) = lim_x→c (sin x / x)

= sin c 

⇒ lim_x→c f(x) = f(c) 

Therefore, f is continuous at all points x, such that x < 0.

Case II:

If c > 0, then f(c) = c + 1

⇒ f(c) = c + 1

lim_x→c f(x) = lim_x→c (x + 1)

= c + 1

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x > 0.

Case III:

If c = 0, then f(c) = f(0) = 0 + 1 = 1

The left hand limit of f at x = 0 is,

lim_x→0⁻ f(x) = lim_x→0⁻ (sinx/x) = 1

The right hand limit of f at x = 0 is,

lim_x→0⁺ f(x) = lim_x→0⁺ (x + 1) = 1

⇒ lim_x→0⁻ f(x) = lim_x→0⁺ f(x) = f(0) 

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at all points of the real line.

Thus, f has no point of discontinuity

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 23

Find the points of discontinuity of f , where f(x) = {(sinx/x, if x < 0) (x+1, if x ≥ 0)

Summary:

For the given function  f(x) = {(sinx/x, if x < 0) (x+1, if x ≥ 0) , it can be concluded that f is continuous at all points of the real line. Thus, f has no point of discontinuity