Question

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

f(x) = `{(x^10 - 1", if"  x<=1),(x^2", if"  x > 1):}`

Answer 1

f(x) = `{(x^10 - 1", if"  x<=1),(x^2", if"  x > 1):}`

For x < 1, f(x) = x¹⁰ − 1 and

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

So this is a function.

At x = 1,

`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (x¹⁰ − 1)

= `lim_(h -> 0)` [(1 − h)¹⁰ − 1]

= (1 − 0)¹⁰ − 1

= 1 − 1

= 0

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

= `lim_(h -> 0)` (1 + h)²

= `lim_(h -> 0)` (1 + h² + 2h)

= 1

Hence, f is not continuous at x = 1.