Question

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

f(x) = `{(3", if"  0 <= x <= 1),(4", if"  1 < x < 3),(5", if"  3 <= x <= 10):}`

Answer 1

f(x) = `{(3", if"  0 <= x <= 1),(4", if"  1 < x < 3),(5", if"  3 <= x <= 10):}`

For 0 ≤ x ≤ 1, f(x) = 3;

1 < x < 3, f(x) = 4 and

3 ≤ x ≤ 10, f(x) = 5, is a continuous function.

So this is a function.

At x = 1,

`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (3) = 3

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

Hence, f is not continuous at x = 1.

At x = 3,

`lim_(x -> 3^-)` f(x) = `lim_(x -> 3^-)` (4) = 4

`lim_(x -> 3^+)` f(x) = `lim_(x -> 3^+)` (5) = 5

Hence, f is not continuous at x = 3.