Find all points of discontinuity of f,where f is defined by f(x)={(2x + 3, if x ≤ 2) (2x − 3, if x > 2)

Solution:

A function is said to be continuous when the graph of the function is a single unbroken curve.

According to the given problem:

The given function is

f(x)={(2x + 3, if x ≤ 2) (2x − 3, if x > 2)

It is evident that the given function f is defined at all the points of the real line.

Let c be a point on the real line.

Then, three cases arise.

c < 2

c > 2

c = 2

Case I: 

c < 2

f(c) = 2c + 3

Then,

lim_x→c f(x) = lim_x→c (2x + 3)

= 2c + 3

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x < 2

Case II:

c > 2

Then,

f(c) = 2c − 3

lim_x→c f(x) = lim_x→c (2x − 3)

= 2c − 3

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x > 2

Case III:

c = 2

Then, the left hand limit of ff at x = 2 is,

lim_x→2⁻ f(x) = lim_x→2⁻ (2x+3)

= 2(2) + 3 = 7

The right hand limit of f at x = 2 is,

lim_x→2⁺ f(x) = lim_x→2⁺ (2x − 3)

= 2(2) − 3 = 1

It is observed that the left and right-hand limits of f at x = 2 do not coincide.

Therefore, f is not continuous at x = 2.

Hence, x = 2 is the only point of discontinuity of f

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 6

Find all points of discontinuity of f,where f is defined by f(x)={(2x + 3, if x ≤ 2) (2x − 3, if x > 2)

Summary:

f is defined by f(x)={(2x + 3, if x ≤ 2) (2x − 3, if x > 2)  hence f is continuous at all points x, such that x < 2,f is continuous at all points x, such that x > 2,f is not continuous at x = 2