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):}`

Answer 1

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.