Discuss the continuity of the function f, where f is defined by f(x) = {(−2, if x ≤ −1) (2x, if −1 < x ≤ 1) (2, if x > 1)

Solution:

The given function is f(x) = {(−2, if x ≤ −1) (2x, if −1 < x ≤ 1) (2, if x > 1)

The given function f is defined at all the points.

Let c be a point on the real line.

Case I:

If c < -1, then f(c) = -2

lim_x→c f(x) = lim_x→c (-2)

= -2

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x < -1.

Case II:

If c = -1, then f(-1) = -2

The left-hand limit of f at x = -1 is,

lim_x→-1⁻ f(x) = lim_x→-1⁻ (-2) = -2

The right hand limit of f at x = -1 is,

lim_x→-1⁺ f(x) = lim_x→-1⁺ (2x)

= 2(-1) = -2

Therefore, f is continuous at x = -1.

Case III:

If -1 < c < 1, then f(c) = 2c

lim_x→c f(x) = lim_x→c (2x) = 2c

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at in the interval (-1,1).

Case IV:

If c = 1, then f(1) = 2(1) = 2

The left-hand limit of f at x = 1 is,

lim_x→1⁻ f(x) = lim_x→1⁻ (2x) = 2(1) = 2

The right-hand limit of f at x = 1 is,

lim_x→1⁺ f(x) = lim_x→1⁺ (2) = 2

Therefore, f is continuous at x = 2.

Case V:

If c > 1, then f(c) = 2

lim_x→c f(x) = lim_x→c(2) = 2

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x such that x > 1.

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

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 16

Discuss the continuity of the function f, where f is defined by f(x) = {(−2, if x ≤ −1) (2x, if −1 < x ≤ 1) (2, if x > 1)

Summary:

For the function f, where f is defined by f(x) = {(−2, if x ≤ −1) (2x, if −1 < x ≤ 1) (2, if x > 1),  f is continuous at all points x such that x > 1.Hence, it can be concluded that f is continuous at all points of the real line