Question

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

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

Answer 1

We observe that f is continuous at all real numbers x < 1 and x > 1.

Now, continuity at x = 1

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

= `lim_(h -> 0)` [(1 − h)² + 1]

= `lim_(h -> 0)` [1 + h² − 2h + 1]

= `lim_(h -> 0)` [2 + h² − 2h]

= 2 + 0 − 0

= 2

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

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

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

= 2 + 0

= 2

f(1) = 1 + 1 = 2

Hence, f is a function at x = 1.

There are no points of discontinuity here.