Question

Find all points of discontinuity of f, where f is defined by `f(x) = {(x^3 - 3, if x <= 2),(x^2 + 1, if x > 2):}`

Answer 1

`f(x) = {(x^3 - 3, if x <= 2),(x^2 + 1, if x > 2):}`

For x < 2, f(x) = x³ - 3 and

x > 2, f(x) = x² + 1 is a polynomial function.

So this is a function.

At x = 2,

`lim_(x -> 2^-)` f(x) = `lim_(x -> 2^-)` (x³ - 3)

`= lim_(h -> 0) [(2 - h)^3 - 3]`

`= lim_(h -> 0) [8 - h^3 - 12 h + 6h^2 - 3]`

`= lim_(h -> 0) (5 - h^3 - 12h + 6h^2) = 5`

`lim_(x -> 2^+) f(x) = lim_(x -> 2^+) (x^2 + 1)`

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

`= lim_(h -> 0) (4 + h^2 + 4h + 1)`

`= lim_(h -> 0) (5 + h^2 + 4h)`

= 5

f(2) = (2)³ - 3

= 8 - 3

= 5

Hence, f is a function at x = 2.

There are no points of discontinuity here.