Question

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(2x + 3", if"  x<=2),(2x - 3", if"  x > 2):}`

Answer 1

f(x) = `{(2x + 3", if"  x<=2),(2x - 3", if"  x > 2):}`

`lim_(x -> 2^-)` f(x) = `lim_(x -> 2^-)` (2x + 3)

= `lim_(h -> 0)` [2(2 − h) + 3]

= `lim_(h -> 0)` [4 − 2h + 3]

= `lim_(h -> 0)` (7 − 2h)

= 7 − 2 × 0

= 7

`lim_(x -> 2^+)` f(x) = `lim_(x -> 2^+)` (2x − 3)

= `lim_(h -> 0)` [2(2 + h) − 3]

= `lim_(h ->0)` [4 + 2h − 3]

= `lim_(h ->0)` (1 + 2h)

= 1 + 2 × 0

= 1

Therefore, f is not continuous at x = 2.