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# 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

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

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