Question

Is the function f defined by f(x) = `{(x", if"  x<=1),(5", if"  x > 1):}`  continuous at x = 0? At x = 1? At x = 2?

Answer 1

f(x) = `{(x", if"  x<=1),(5", if"  x > 1):}`

(i) At x = 0

`lim_(x -> 0^-)` f(x) = `lim_(h -> 0)` f(0 − h)

= 0 − 0

= 0

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

= 0 + 0

= 0

f(0) = 0

Hence, f is continuous at x = 0.

(ii) At x = 1

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

= 1 − 0

= 1

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

= 5

f(1) = 1

Hence, f is not continuous at x = 1.

(iii) At x = 2

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

= 5

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

= 5

f(2) = 5

Hence, f is continuous at x = 2.