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):}`

Answer 1

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.