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Discuss the continuity of the function f, where f is defined by: f(x) = {2⁢𝑥, if 𝑥 < 0, 0, if 0 ≤ 𝑥 ≤ 1, 4⁢𝑥, if 𝑥 > 1 - Mathematics

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Question

Discuss the continuity of the function f, where f is defined by:

f(x) = `{(2x", if"  x < 0),(0", if"  0 <= x <= 1),(4x", if"  x > 1):}`

Sum
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Solution

f(x) = `{(2x", if"  x < 0),(0", if"  0 <= x <= 1),(4x", if"  x > 1):}`

For x < 0, f(x) = 2x;

0 < x < 1, f(x) = 0 and

for x > 1, f(x) = 4x is a polynomial and continuous function.

So this is a function.

At x = 0,

`lim_(x -> 0^-)` f(x) = `lim_(x -> 0^-)` (2x)

= `lim_(h -> 0)` [2(0 − h)]

= `lim_(h -> 0)` (−2h)

= −2 × 0

= 0

`lim_(x -> 0^+)` f(x) = `lim_(x -> 0^+)` (0) = 0

Hence, f is continuous at x = 0.

At x = 1,

`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (0) = 0

`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (4x)

= `lim_(h -> 0)` [4(1 + h)]

= `lim_(h -> 0)` (4 + 4h)

= 4 + 4 × 0

= 4

Hence, it is not continuous at x = 1.

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Chapter 5: Continuity and Differentiability - Exercise 5.1 [Page 160]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 5 Continuity and Differentiability
Exercise 5.1 | Q 15 | Page 160

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