Question

How many point of discontinuity for the following function for x ∈ R

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

Options

• 4

• 3

• 1

• 0

Answer 1

0

Explanation:

At `x` = 1, L.H.L = `lim_(x -> 1) f(x) = lim_(x -> 1) (x^2 + 1)` = 2

R.H.L = `lim_(x -> 1^+) f(x) = lim_(x -> 1^+) (x + 1)` = 2

`f(1)` = 1 + 1 = 2

⇒ `f` is continuous at `x` = C > 1

At `x` = C > 1,

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

⇒ `f` is continuous at `x` = C > 1

At `x` = C > 1,

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

⇒ `f` is continuous at `x` = C < 1

There is no point of discontinuity at any point x ∈ R.