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Question
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):}`
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Solution
f(x) = `{(-2", if" x <= -1),(2x", if" -1 < x <= 1),(2", if" x > 1):}`
f(x) = −2 for x < 1;
−1 < x < 1, f(x) = 2x and
x > 1, f(x) = 2 is a polynomial function.
So this is a function.
At x = −1,
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (−2) = −2
`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (2x)
= `lim_(h -> 0)` [2 (−1 + h)]
= `lim_(h -> 0)` (−2 + 2h)
= −2 + 0
= −2
f(−1) = −2
Hence, f is continuous at x = −1.
At x = 1,
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (2x)
= `lim_(h -> 0)` [2(1 − h)]
= `lim_(h -> 0)` (2 − 2h)
= 2 − 2 × 0
= 2
`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (2) = 2
f(1) = 2 × 1 = 2
Hence, f is continuous at x = 1.
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