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 − h³ − 12h + 6h² − 3]

= `lim_(h -> 0)` (5 − h³ − 12h + 6h²)

= 5

`lim_(x -> 2^+)` f(x) = `lim_(x -> 2^+)` (x² + 1)

= `lim_(h -> 0)` [(2 + h)² + 1]

= `lim_(h -> 0)` (4 + h² + 4h + 1)

= `lim_(h -> 0)` (5 + h² + 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.